Latest revision as of 05:14, 19 December 2021

1 Introducción

En optimización, la estrategia de búsqueda en línea es uno de los enfoques iterativos básicos para encontrar un mínimo local Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \displaystyle v^{*}}

de una función objetivo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \displaystyle f:\mathbb {R} ^{n}\to \mathbb {R}}

. El enfoque de búsqueda en línea primero supone conocida una aproximación previa Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}

para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v^*}
y encuentra una dirección de descenso Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
a lo largo de la cual la función objetivo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \displaystyle f}
 será minimizada y entonces esto  determinará qué tan lejos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}
 debe moverse a lo largo de esa dirección. Este tamaño de paso está asociado con un mínimo local de la restricción de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}
a la recta de búsqueda. Al problema de encontrar este tamaño de paso se le llama un problema de búsqueda en línea, es decir, es un problema de minimización con  restricciones de la forma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{\begin{array}{c}Min\hbox{ }f(v) \\ \hbox{sujeto a } v =u -\rho w , \forall \rho \in \mathbb{R};u ,w \in \mathbb{R}^{n} ,\end{array}\right.

que también se puede escribir como

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Min_{\rho \in \mathbb{R}}\hbox{ }f(u -\rho w) .

(1)

Aquí la recta de búsqueda es la recta que pasa por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}

y tiene (por conveniencia, para nuestras posteriores aplicaciones) la dirección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -w}

. Como ya se dijo, una de las áreas donde surgen problemas de búsqueda en línea son los métodos iterativos para minimizar localmente una función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f :\mathbb{R}^{n} \rightarrow \mathbb{R}}

sin restricciones. En cada iteración Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}
debe resolverse un problema de búsqueda en línea de la forma (1), para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
conocidos. Si denotamos con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _{i}}
a la solución del problema de búsqueda en línea de la iteración Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle i}

, se toma a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v_{i} =u -\rho _{i}w}

como la nueva aproximación para el mínimo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}

. La solución de los problemas de búsqueda en línea se puede aproximar por una variedad de métodos numéricos [1], entre ellos el método de Newton. Si definimos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g(\rho )=f(u -\rho w)}

entonces la solución del problema (1) es equivalente a la solución del problema en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Min_{\rho \in \mathbb{R}}\hbox{ }g(\rho ) .

Para resolver este problema buscamos los valores donde la derivada de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g}

se hace cero, es decir, resolvemos (por el método de Newton) la ecuación

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g'(\rho ^{*})=0 .

Tenemos que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g'(\rho )=-Df(u-\rho w;w)=-\nabla f(u-\rho w)\cdot w ,

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle Df(v;w)}

 denota la derivada direccional de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v}
en la dirección de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}

. Así que por comodidad definimos

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H(\rho )=-g'(\rho )=Df(u-\rho w;w)=\nabla f(u-\rho w)\cdot w

y resolvemos la ecuación

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): H(\rho ^{*})=0,

para lo cual, dada una aproximación inicial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _{0}} , se construyen sucesivas aproximaciones de acuerdo a

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _{i +1} =\rho _{i} -\frac{H(\rho _{i})}{H^{ \prime }(\rho _{i})} ,i =0 ,1 ,2 , . . . .

En este trabajo extenderemos estas ideas para aplicar el método de Newton a un problema de búsqueda en línea pero donde los elementos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}
aunque conocidos, pertenecen al espacio de Hilbert Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (L^{2})^3}
y no a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb {R} ^{n}}

. En este contexto, este artículo es una continuación o extensión de lo expuesto en [2].

2 Descripción del problema

Los problemas de búsqueda en línea en que estamos interesados aquí son del tipo

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{c}\hbox{Encontrar }\mathbf{\rho ^*}\in \mathbb{R},\hbox{ tal que } \\ J(\mathbf{u}-\rho ^* \mathbf{w})\leq J(\mathbf{u}-\rho \mathbf{w}),\forall \rho \in \mathbb{R},\end{array}\right.

(2)

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}}
fijos en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (L^{2} (0,T))^3}

, y donde el funcional de nuestro interés Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J:(L^{2} (0,T))^3 \longrightarrow \mathbb {R}}

es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): J(\mathbf{v})=\frac{\eta }{2}\int \limits _{0}^{T}\Vert \mathbf{v} \Vert ^{2} dt+\frac{k}{2}||\mathbf{y}(T)-\mathbf{y}_{T}||^{2},

(3)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k,\eta >0}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle ||.||}
la norma euclideana canónica, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}}
es una función vectorial, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}=(v_1(t),v_2(t), v_3(t))^T}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \mathbf{v}\Vert ^2=v_1^2+v_2^2+v_3^2}

y la función vectorial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}=( y_{1},y_{2},y_{3})^T }
es la solución del siguiente problema de valor inicial que modela la dinámica de un circuito de tres juntas de Josephson acopladas inductivamente, [3]:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{c}\gamma _{1}\frac{dy_{1}}{dt}+\kappa _{1}(y_{1}-y_{2})+\sen y_{1}=i_{1}+v_{1},\hbox{ en }(0,T), \\ \gamma _{2}\frac{dy_{2}}{dt}+\kappa _{1}(y_{2}-y_{1})+\kappa _{2}(y_{2}-y_{3})+\sen y_{2}=i_{2}+v_{2},\hbox{ en }(0,T), \\ \gamma _{3}\frac{dy_{3}}{dt}+\kappa _{2}(y_{3}-y_{2})+\sen y_{3}=i_{3}+v_{3},\hbox{ en }(0,T), \\ \mathbf{y}(0)=\mathbf{y}_{0}.\end{array}\right.

(4)

En (3)-(4), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}_{0}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}_{T}}
son estados inicial y final conocidos, respectivamente. En [3] y en [4] se dan los siguientes valores factibles para los parámetros involucrados en este sistema y con los cuales se harán los experimentos numéricos mas adelante:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \gamma _{1} =0.7,\hbox{ }\gamma _{2}=1.1,\hbox{ }\gamma _{3}=0.7,\hbox{ }i_{1}=1,\hbox{ }i_{2}=0.8,\hbox{ }i_{3}=-1,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \hbox{ }\kappa _{1} =\kappa _{2}=0.1.

2.1 El problema de minimización global y el diferencial de J

El problema de minimización global o sin restricciones asociado con los problemas de búsqueda en línea descritos es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{c}\hbox{Encontrar }v^{*}\in (L^{2}(0,T))^{3}, \hbox{tal que }\\ J(v^{*})\leq J(v),\forall v\in (L^{2}(0,T))^{3},\end{array}\right.

el cual, (junto con (3)-(4)) corresponde a un problema de control cuyo objetivo es llevar la dinámica del sistema del estado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}_{0}}

al estado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}_{T}}

. Este tipo de problemas de control pueden resolverse usando un algoritmo de gradiente conjugado como los discutidos en el Capítulo 2 de [5] y el Capítulo 3 de [6], donde también se menciona el concepto de Frechet-diferenciabilidad adecuado para minimizar funcionales en espacios de Hilbert. A continuación damos una definción y un teorema básicos para resolver el problema que nos ocupa.

Definición. Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}

un espacio de Hilbert. Un funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}
sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}
es Frechet-diferenciable si, para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v,w \in V}

, existe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle DJ(v) \in V'} , la derivada o el diferencial de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v}

, tal que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): J(v+w)-J(v)=\langle DJ(v),w\rangle + \Vert w \Vert \epsilon (v,w),

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \langle . , . \rangle }

denotando par de dualidad, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon (v,w)}
tendiendo a cero cuando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert w \Vert }
tiende a cero.

Teorema. Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}

está definido como en (3) entonces  es Frechet-diferenciable y su diferencial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle DJ(\mathbf{v})}
es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): DJ(\mathbf{v})=\eta \mathbf{v}+\mathbf{p},

(5)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{p}}

se obtiene al resolver el sistema (versión matricial de (4)) :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}\Gamma \dfrac{d\mathbf{y}}{dt}+K\mathbf{y}+ \sen \mathbf{y}=\left( \begin{array}{ccc}i_1+v_1 \\ i_2+v_2 \\ i_3+v_3 \\ \end{array} \right)\hbox{ en } (0,T), \\ \mathbf{y}(0)=\mathbf{y}_0, \end{array} \right.

(6)

y después el sistema adjunto

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}-\Gamma \dfrac{d\mathbf{p}}{dt}+K\mathbf{p}+ C(\mathbf{y}) \; \mathbf{p} =0 \hbox{ en } (0,T), \\ \Gamma \mathbf{p}(T)=k(\mathbf{y}(T)-\mathbf{y}_{T}), \end{array} \right.

(7)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C(\mathbf{y})}

es una matriz diagonal de 3x3 para la cual la entrada iésima de la diagonal es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle cos(y_i)}

.

De este teorema resulta que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle DJ(\mathbf{u}-\rho \mathbf{w})=\eta (\mathbf{u}-\rho \mathbf{w})+\mathbf{p}} , con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{p}}

solución de (7) una vez que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}(t)}
 es solución del sistema (6) tomando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}=\mathbf{u}-\rho \mathbf{w}}

.

La demostración de este teorema, y de los resultados que se mencionan a continuación pueden consultarse en [7].

3 Metodología de solución

Para resolver por el método de Newton nuestro problema de búsqueda en línea (2) definimos

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g(\rho )=J(\mathbf{u}-\rho \mathbf{w}),

(8)

entonces el problema se reduce a encontrar raíces para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g'}

(un problema de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}}

), las cuales aproximaremos con la iteración de Newton

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _{i+1}=\rho _i-\dfrac{g'(\rho _i)}{g''(\rho _i)} ; g''(\rho _i)\neq 0,

para lo cual debemos conocer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g'}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g''}

. Para describir Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g'(\rho )}

en términos de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle DJ}
necesitamos el siguiente teorema:

Teorema. Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}

un funcional Frechet-diferenciable sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g}
como en (8). Entonces

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g'(\rho )=\dfrac{d}{d\rho }J(\mathbf{u}-\rho \mathbf{w})=-\langle DJ (\mathbf{u}-\rho \mathbf{w}), \mathbf{w} \rangle{.}

Así que la ecuación que tenemos que resolver para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho }

es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g'(\rho )=-\langle DJ (\mathbf{u}-\rho \mathbf{w}), \mathbf{w} \rangle=0.

Como la iteración de Newton es

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _{i+1}=\rho _{i}-\dfrac{g'(\rho _{i})}{g''(\rho _{i})},

aún nos falta encontrar una expresión para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g''(\rho )} . Como estamos trabajando en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (L^2(0,T))^3} , los pares de dualidad coinciden con el producto interno gracias al teorema de representación de Riesz. Dado que ya conocemos el representante de la transformación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle DJ(v)}

(ver (5)), podemos escribir

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g'(\rho )=-(\eta (\mathbf{u}-\rho \mathbf{w})+\mathbf{p}, \mathbf{w})=-\int _{0}^{T}(\eta (\mathbf{u}-\rho \mathbf{w})+\mathbf{p})\cdot \mathbf{w} \hbox{ dt},

(9)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{p}}

se obtiene resolviendo

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{c}\gamma _{1}\frac{d y_{1\rho }}{dt}+\kappa _{1}( y_{1\rho }-y_{2\rho })+ \sen y_{1\rho }=i_1+u_1-\rho w_1\hbox{ en }(0,T), \\ \gamma _{2}\frac{d y_{2\rho }}{dt}+\kappa _{1}( y_{2\rho }- y_{1\rho })+\kappa _{2}( y_{2\rho }- y_{3\rho })+\sen y_{2\rho }=i_2+ u_2-\rho w_2 \hbox{ en }(0,T), \\ \gamma _{3}\frac{d y_{3\rho }}{dt}+\kappa _{2}( y_{3\rho }- y_{2\rho })+ \sen y_{3\rho }=i_3+u_3-\rho w_3\hbox{ en }(0,T), \\ \mathbf{y}(0)=\mathbf{y}_0.\end{array}\right.

y luego el problema

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}-\Gamma \dfrac{d\mathbf{p}_{\rho }}{dt}+K\mathbf{p}_{\rho }+ \left( \begin{array}{ccc}\cos y_{1 \rho } & 0 & 0 \\ 0 & \cos y_{2 \rho } & 0 \\ 0 & 0 & \cos y_{3 \rho } \\ \end{array} \right)\mathbf{p}_{\rho } =0 \hbox{ en } (0,T), \\ \Gamma \mathbf{p}_{\rho }(T)=k(\mathbf{y}_{\rho }(T)-\mathbf{y}_{T}). \end{array} \right.

Para calcular Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g''(\rho )=\dfrac{dg'}{d\rho }} , por la regla de Leibniz para la diferenciación bajo el signo integral tenemos, a partir de (9) que:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{dg'(\rho )}{d\rho }=-\int _{0}^{T} \dfrac{\partial }{\partial \rho } (\eta (\mathbf{u}-\rho \mathbf{w})+\mathbf{p})\cdot \mathbf{w} \hbox{ dt },

lo cual nos da

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g''(\rho )=\int _{0}^{T} \left[\eta \mathbf{w}-\dot{\mathbf{p}}_{\rho } \right]\cdot \mathbf{w}\hbox{ dt },

que denotaremos como

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g''(\rho )=\langle D^{2}J (\mathbf{u}-\rho \mathbf{w})\mathbf{w}, \mathbf{w} \rangle

y donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dot{\mathbf{p}}_{\rho }}

se obtiene resolviendo

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}\Gamma \dfrac{d\dot{\mathbf{y}}_{\rho }}{dt}+K\dot{\mathbf{y}}_{\rho }+ \left( \begin{array}{ccc}\cos y_{1 \rho } & 0 & 0 \\ 0 & \cos y_{2 \rho } & 0 \\ 0 & 0 & \cos y_{3 \rho } \\ \end{array} \right)\dot{\mathbf{y}}_{\rho }= \left( \begin{array}{ccc}-w_1 \\ -w_2 \\ -w_3 \\ \end{array} \right) \hbox{ en } (0,T), \\ \dot{\mathbf{y}}_{\rho }(0)=0 \end{array} \right.

y luego el problema

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}-\Gamma \dfrac{d\dot{\mathbf{p}}_{\rho }}{dt}+K\dot{\mathbf{p}}_{\rho }+ \left( \begin{array}{ccc}\cos y_{1 \rho } & 0 & 0 \\ 0 & \cos y_{2 \rho } & 0 \\ 0 & 0 & \cos y_{3 \rho } \\ \end{array} \right)\dot{\mathbf{p}}_{\rho }= \\ \left( \begin{array}{ccc}\sen y_{1 \rho } \; \dot{y}_{1 \rho } & 0 & 0 \\ 0 & \sen y_{2 \rho } \; \dot{y}_{2 \rho } & 0 \\ 0 & 0 & \sen y_{3 \rho } \; \dot{y}_{3 \rho } \\ \end{array} \right)\mathbf{p}_{\rho } \hbox{ en } (0,T), \\ \Gamma \dot{\mathbf{p}}_{\rho }(T)=k\dot{\mathbf{y}}_{\rho }(T). \end{array} \right.

Con ésto el método de Newton para aproximar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho ^{*}}

queda

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _{i+1}=\rho _{i}-\dfrac{g'(\rho _{i})}{g''(\rho _{i})}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _{i+1}=\rho _{i}-\dfrac{-\langle DJ (\mathbf{u}-\rho \mathbf{w}), \mathbf{w} \rangle }{\langle D^{2}J (\mathbf{u}-\rho \mathbf{w})\mathbf{w}, \mathbf{w} \rangle }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho _{i+1}=\rho _{i}+\dfrac{\langle DJ (\mathbf{u}-\rho \mathbf{w}), \mathbf{w} \rangle }{\langle D^{2}J (\mathbf{u}-\rho \mathbf{w})\mathbf{w}, \mathbf{w} \rangle }.

4 Discretización del problema de búsqueda en línea

La idea principal es sustituir el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^2(0,T)}

por un espacio de funciones más práctico. A grandes rasgos este espacio será el de las funciones lineales por pedazos en (0,T), que además ofrecen la ventaja de poder ser representadas computacionalmente por eneadas de números.

Para describir esto con mayor precisión, comenzamos haciendo una partición uniforme del intervalo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (0,T)}

con

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}

número de subintervalos de la partición uniforme.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=\frac{T}{N}}

 tamaño de cada subintervalo.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle tt}

un vector de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N+1}
puntos espaciados uniformemente en el intervalo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (0,T)}
de la forma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): tt=[0,\cdots ,tt_{N}].

Con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle tt_i=i*h, i=0,1,2,\cdots ,N} , esto significa que en el arreglo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle tt}

se almacena la partición en el tiempo.

Dada esta partición, una función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f(t)}

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^2(0,T)}
puede ser aproximada por la función lineal por pedazos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_h(t)}
cuyo valor en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_i}
 es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_h(t_i)=f(t_i)}

. En la figura 1 se muestra una Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}

y su aproximación lineal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f_h}
por pedazos, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N=10}

. Note que entonces una función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}

puede ser representada computacionalmente por dos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (N+1)}

-eadas:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f\approx \left\lbrace \begin{array}{c} (tt_0=0,\cdots , tti,\cdots ,tt_N) \\ (f(0),\cdots ,f(tt_i),\cdots ,f(tt_N)). \\ \end{array} \right.

$Función f(t)=t \exp (-t) y fₕ(tt)=tt \exp (-tt), con N=10, T=5 y h=0.5.$

Figura 1: Función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f(t)=t \exp (-t)

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f_h(tt)=tt \exp (-tt),
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): N=10, T=5
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h=0.5

.

Con esto, está claro que podemos aproximar las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}}

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (L^2(0,T))^3}
 por triadas de funciones lineales por pedazos, es decir por matrices de la forma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}(t)\approx (v_1,v_2,v_3)(tt)=\left( \begin{array}{ccc} v_1(0)&\cdots &v_1(tt_{N}) \\ v_2(0)&\cdots &v_2(tt_{N}) \\ v_3(0)&\cdots &v_3(tt_{N}) \\ \end{array} \right),}

mientras que las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}}

de nuestra teoría quedarían aproximada o representadas por la matriz

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{u}=\left( \begin{array}{ccc} u_1(0)&\cdots &u_1(tt_N) \\ u_2(0)&\cdots &u_2(tt_N) \\ u_3(0)&\cdots &u_3(tt_N) \\ \end{array} \right),

y las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}}

de nuestra teoría quedarían aproximadas o representadas por la matriz

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{w}=\left( \begin{array}{ccc} w_1(0)&\cdots &w_1(tt_N) \\ w_2(0)&\cdots &w_2(tt_N) \\ w_3(0)&\cdots &w_3(tt_N) \\ \end{array} \right).

También a partir de ahora, estaremos usando la notación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}^n}

para denotar el valor de la función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}}

, lineal por pedazos, en el tiempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle nh} . Esta notación también aplica para las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{p}}

, es decir, con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}^n}

denotamos el valor de la función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}}
en el tiempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle nh}
y similarmente para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{p}}

. Esto se resume en decir que en este capítulo damos los valores de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}}
en la partición Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle tt}
y debemos minimizar una version discreta del funcional, para lo cual es necesario encontrar los valores aproximados  de

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{y}\approx (y_1,y_2,y_3)(tt)=\left( \begin{array}{ccc} y_1(0)&\cdots &y_1(tt_N) \\ y_2(0)&\cdots &y_2(tt_N) \\ y_3(0)&\cdots &y_3(tt_N) \\ \end{array} \right),

y de

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{p}\approx (p_1,p_2,y_3)(tt)=\left( \begin{array}{ccc} p_1(0)&\cdots &p_1(tt_N) \\ p_2(0)&\cdots &p_2(tt_N) \\ p_3(0)&\cdots &p_3(tt_N) \\ \end{array} \right)

en la partición Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle tt} . En lo que sigue detallamos cómo hacemos todas estas discretizaciones.

4.1 Discretización del Funcional

El funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}

de (3) lo aproximamos por

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): J^{h}(\mathbf{v})=\frac{\eta h}{2}\sum _{n=1}^{N} ||\mathbf{v}^n ||^{2}+\frac{k}{2}||\mathbf{y}^N-\mathbf{y}_T||^{2},

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle || \mathbf{v}^{n} ||^{2}=| v_{1}^{n} |^{2}+| v_{2}^{n} |^{2}+| v_{3}^{n}|^{2}} . Aquí Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}^N}

es la aproximación de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}}
en el tiempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (tt_N=Nh)}

, que mas adelante especificamos cómo calcular.

4.2 Discretización del SEDO

Aproximamos el sistema (6) por un esquema de Euler explícito que luce como sigue:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}\hbox{Dado } \mathbf{y}^0=\mathbf{y}_0, \\ \\ \hbox{ para } n=0,...,N-1 \hbox{ resolver } \\ \\ \mathbf{y}^{n+1}= \mathbf{y}^{n}-h \Gamma ^{-1} K \mathbf{y}^{n}- h \Gamma ^{-1} \left( \begin{array}{ccc}\sen y_1^{n} \\ \sen y_2^{n} \\ \sen y_3^{n} \\ \end{array} \right)+ h \Gamma ^{-1} \left( \begin{array}{ccc}i_1+v_{1}^{n} \\ i_2+v_{2}^{n} \\ i_3+v_{3}^{n} \end{array} \right). \end{array} \right.

4.3 Discretización del SEDO Adjunto

El sistema adjunto (7) lo aproximamos por

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}\mathbf{p}^{N}=k \Gamma ^{-1} (\mathbf{y}^{N}-\mathbf{y}_T), \\ \\ \hbox{ para } n=N-1,...,0 \hbox{ resolver } \\ \\ \mathbf{p}^{n}=\mathbf{p}^{n+1}-h \Gamma ^{-1} K \mathbf{p}^{n+1}-h \Gamma ^{-1} \left( \begin{array}{ccc}\cos y_{1}^{n} & 0 & 0 \\ 0 & \cos y_{2}^{n} & 0 \\ 0 & 0 & \cos y_{3}^{n} \\ \end{array} \right)\; \mathbf{p}^{n+1}. \\ \end{array} \right.

4.4 Discretización del Producto interno

Las discretizaciones mencionadas implican que estamos considerando el siguiente producto interno

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): (\mathbf{u},\mathbf{w})_{h}=h \sum _{n=1}^N \mathbf{u}^n \cdot \mathbf{w}^n, \forall \mathbf{u}, \mathbf{w} \in (\mathbb{R}^3)^N.

4.5 Discretización de g'(ρ) y g(ρ)

Para resolver la ecuación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g(\rho ^*)=0}

con el método de Newton se requiere el conocimiento de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g'(\rho )}
que estará dada en los nuevos espacios vectoriales por

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g'(\rho )=-h \eta \sum _{n=1}^{N}[\mathbf{u}^{n}-\rho \mathbf{w}^{n}+\mathbf{p}_{\rho }^{ n}]\cdot \mathbf{w}^{n},

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{p}_{\rho }^{n}}

se obtiene resolviendo

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}\hbox{Dado } \mathbf{y}_{\rho }^0=\mathbf{y}_{\rho }(0), \\ \\ \hbox{ para } n=0,...,N-1 \hbox{ resolver } \\ \\ \mathbf{y}_{\rho }^{n+1}= \mathbf{y}_{\rho }^{n}-h\Gamma ^{-1} K \mathbf{y}_{\rho }^{n}-h\Gamma ^{-1} \left( \begin{array}{ccc}\sen y_{\rho{1}}^{n} \\ \sen y_{\rho{2}}^{n} \\ \sen y_{\rho{3}}^{n} \\ \end{array} \right) +h\Gamma ^{-1} \left( \begin{array}{ccc}i_1+u_{1}^{n}-\rho w_{1}^{n} \\ i_2+u_{2}^{n} -\rho w_{2}^{n} \\ i_3+u_{3}^{n} -\rho w_{3}^{n} \\ \end{array} \right). \\ \end{array} \right.

y luego resolviendo

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\lbrace \begin{array}{c}\mathbf{p}_{\rho }^{N}=k\Gamma ^{-1}(\mathbf{y}_{\rho }^{N}-\mathbf{y}_T), \\ \\ \hbox{ para } n=N-1,...,0 \hbox{ resolver } \\ \\ \mathbf{p}_{\rho }^{n}=\mathbf{p}_{\rho }^{n+1}-h\Gamma ^{-1} K \mathbf{p}_{\rho }^{n+1} -h\Gamma ^{-1} \left( \begin{array}{ccc}\cos y_{\rho{1}}^{n+1} & 0 & 0 \\ 0 & \cos y_{\rho{2}}^{n+1} & 0 \\ 0 & 0 & \cos y_{\rho{3}}^{n+1}\\ \end{array} \right)\; \mathbf{p}_{\rho }^{n+1}. \\ \end{array} \right.

y para

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g''(\rho )=h\sum _{n=1}^{N}[\eta \mathbf{w}^{n}-\overset{\cdot }{\mathbf{p}}_{\rho }^{n}] \cdot \mathbf{w}^{n},

con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overset{\cdot }{\mathbf{p}}_{\rho }^{n},}

obtenido al resolver

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{c}\overset{\cdot }{\mathbf{y}}_{\rho }^{0}=\mathbf{0}, \\ \hbox{para } n=0,....,N-1 \hbox{ resolver} \\ \overset{\cdot }{\mathbf{y}}_{\rho }^{n+1}=\overset{\cdot }{\mathbf{y}^{n}}_{\rho }-h \Gamma ^{-1} K \overset{\cdot }{\mathbf{y}}_{\rho }{ n}-h\Gamma ^{-1}\left( \begin{array}{c}\cos y_{1\rho }^{n} \\ \cos y_{2\rho }^{n} \\ \cos y_{3\rho }^{n}\end{array}\right)\overset{\cdot }{\mathbf{y}^{n}}_{\rho }-h\Gamma ^{-1}\left( \begin{array}{c}w_{1}^{n} \\ w_{2}^{n}\\ w_{3}^{n}\end{array}\right),\end{array}\right.

y luego resolver

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\{ \begin{array}{c}\overset{\cdot }{\mathbf{p}}_{\rho }^{N}=k\Gamma ^{-1}\overset{\cdot }{\mathbf{y}}_{\rho }^{N}, \\ \hbox{para }n=N-1,....,0 \hbox{ resolver} \\ \overset{\cdot }{\mathbf{p}}_{\rho }^{n}=\overset{\cdot }{\mathbf{p}}_{\rho }^{n+1}+h \Gamma ^{-1} K \overset{\cdot }{\mathbf{p}}_{\rho }^{n+1} \\ \\ +h \Gamma ^{-1} \left( \begin{array}{ccc}\cos y_{1\rho }^{n}& 0 & 0 \\ 0 & \cos y_{2\rho }^{n} & 0 \\ 0 & 0 & \cos y_{3\rho }^{n}\end{array}\right)\overset{\cdot }{\mathbf{p}}_{\rho }^{n+1} \\ \\ -\left( \begin{array}{ccc}\overset{\cdot }{y}_{1\rho }^{n}\sen y_{1\rho }^{n} & 0 & 0 \\ 0 & \overset{\cdot }{y}_{2\rho }^{n}\sen y_{2\rho }^{n} & 0 \\ 0 & 0 & \overset{\cdot }{y}_{3\rho }^{n}\sen y_{3\rho }^{n}\end{array}\right)\mathbf{p}_{\rho }^{n+1}\mathbf{.}\end{array}\right.

Con esto podemos ya aplicar el método de Newton para resolver la versión discreta de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g'(\rho )=0}

Dado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho ^{0}}

, iterar con

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho ^{m+1}=\rho ^{m}-\frac{g'(\rho ^{m})}{g''(\rho ^{m})}.

Usamos el criterio de paro Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \dfrac{\vert \rho ^{m+1}-\rho ^{m}\vert }{\vert \rho ^{m+1}\vert }< \varepsilon } , para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varepsilon }

pequeño dado.

5 Experimentación computacional

5.1 Ejemplo 1

En este ejemplo tomamos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_1(t)=t \exp (-t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_2(t)=t^3} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_3(t)=0}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_1(t)=\exp (-t)/10}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_2(t)=3t-t^3}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_3(t)=0}

. Se tomaron los valores siguientes para los parámetros en la iteración de Newton y en la discretización de los problemas involucrados:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T=16}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=1.0e-3}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho ^0=0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=T/500}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta=1}

.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}_{0}=[ 1.2514; 0.7456; -0.9753]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}_{T}=[ 7.4207; 6.4958; -0.3236]}

.

Dadas las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}=(t \exp (-t),t^3,0)}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}=(\exp (-t)/10,3t-t^3,0)}

, minimizamos con el método de Newton el funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J(\mathbf{u}(t)-\rho \mathbf{w}(t))}

sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho }
y en 3 iteraciones se obtuvo el valor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho ^{*}=-1.017}

. La gráfica del funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}

restringido a la recta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}=\mathbf{u}-\rho \mathbf{w}}
se muestra en la Figura 2.

Figura 2: Funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): J(\mathbf{u}(t)-\rho \mathbf{w}(t))

para las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{u}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{w}
del ejemplo 1.

5.2 Ejemplo 2

En este ejemplo tomamos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_1(t)=-(t-2)\exp (-t)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_2(t)=3t^2+1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u_3(t)=(t-1)^2+1}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_1(t)=t^3+t-1}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_2(t)=\dfrac{1}{3}t^3-t} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w_3(t)=\exp (-t)/10} . Tanto para la discretización de los problemas involucrados como para las iteraciones de Newton se tomaron los valores siguientes:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle T=12}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=1.0e-3}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho ^0=0.5}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h=T/500}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta=1}

.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}_{0}=[0.1992; 0.1187; -0.1552]}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{y}_{T}=[0.1810; 0.0338; -0.0515]}

.

Dadas las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}=(-(t-2)\exp (-t),3t^2+1,(t-1)^2+1)}

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}=(t^3+t-1,\frac{1}{3}t^3-t,\exp (-t)/10)}

, minimizamos con el método de Newton el funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J(\mathbf{u}(t)-\rho \mathbf{w}(t))}

sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho }
y en 6 iteraciones se obtuvo el valor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho ^{*}=0.084}

. La gráfica del funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}

restringido a la recta Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{v}=\mathbf{u}-\rho \mathbf{w}}
se muestra en la Figura 3.

Figura 3: Funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): J(\mathbf{u}(t)-\rho \mathbf{w}(t))

para las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{u}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \mathbf{w}
del ejemplo 2.

6 Conclusiones

No contamos con un ejemplo para el cual se conozca la solución exacta, así que nuestra validación se basa en el desempeño del método en ejemplos como los dos mostrados, dado que es posible tener una visualización suficientemente detallada del funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J}

restringido a la "recta" que pasa por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{u}}
y tiene dirección Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{w}}

. De acuerdo a estos ejemplos se puede concluir que el método de Newton produce resultados excelentes. En este trabajo el objetivo no es comparar el método de Newton con otros métodos, sino verificar que este tipo de problemas se pueden resolver aceptablemente utilizando el método de Newton; sin embargo en trabajos futuros esperamos considerar otros métodos y hacer tal comparación, por ejemplo, con los métodos que no utilizan derivadas. También podemos mencionar que los resultados presentados aquí se aplicarán próximamente para resolver el problema de minimización global asociado con el problema de control, el cual, como se conocen el estado inicial del que se parte y el estado final al que se quiere llegar, podrá considerarse como un caso de validación de la metodología expuesta en el presente trabajo.

BIBLIOGRAFÍA

[1] R. P. Brent (2002) Algorithms for minimization without derivatives. New York: Courier Dover Publications.

[2] J. López, L. H. Juárez (2019) Método de Newton para Problemas de Búsqueda en Línea en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^2} . Journal of Basic Sciences. Vol. 5, Num. 13.

[3] Y. Braiman, B. Neschke, N. Nair, N. Ima, and R. Glowinski. (2016) Memory States in Small Arrays of Josephson Junctions, PHYSICAL REVIEW E (94), 052223: 1-13.

[4] J. D. Rezac, N. Imam and Y. Braiman, (2017) Parameter optimization for transitions between memory states in small arrays of Josephson junctions, PHYSICA A (474), 267-281.

[5] R. Glowinski (2015) Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems, Philadelphia: SIAM.

[6] R. Glowinski (2003) Finite element methods for incompressible viscous flow. In Handbook of Numerical Analysis, Vol. IX, P.G. Ciarlet & J.L. Lions, eds., 3-1176, Amsterdam: Elsevier.

[7] C. N. Cortazar (2020) Método de Newton para búsquedas en línea en el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (L^2)^3} . Tesis de Maestría. Universidad Juárez Autónoma de Tabasco.