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Zbl 0684.49006
Buttazzo, G.; Cavazzuti, E.
Limit problems in optimal control theory. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, Suppl., 151-160 (1989). ISSN 0294-1449

The authors deal with sequences of optimal control problems of the form $$ ({\frak P}\sb h)\quad \min \{\int\sp{1}\sb{0}f\sb h(t,y,u)dt:\quad y'=g\sb h(t,y,u),\quad y(0)=y\sp 0\sb h\}, $$ where the state variable y belongs to the Sobolev space $Y=W\sp{1,1}(0,1;{\bbfR}\sp n)$ and the control variable u is in $U=L\sp 1(0,1;{\bbfR}\sp m)$. A ``limit problem'' (${\frak P}\sb{\infty})$ is constructed such that - if $(u\sb h,y\sb h)$ is an optimal pair for (${\frak P}\sb h)$ and if $(u\sb h,y\sb h)$ tends to $(u\sb{\infty},y\sb{\infty})$ in the topology $wL\sp 1(0,1;{\bbfR}\sp m)\times L\sp{\infty}(0,1;{\bbfR}\sp n)$, then $(u\sb{\infty},y\sb{\infty})$ is an optimal pair for (${\frak P}\sb{\infty}).$ \par It is shown that, when the functions $g\sb h(t,y,u)$ appearing in the state equations are rapidly oscillating, it may arrive that the domain of the limit problem (${\frak P}\sb{\infty})$ is not given by a state equation $y'=g\sb{\infty}(t,y,u)$, and, in some situations, it may coincide with the whole product space $U\times Y.$ \par The basic tool used for constructing the limit problem (${\frak P}\sb{\infty})$ is the $\Gamma$-convergence theory for functionals defined on a product space.
[G.Buttazzo]

MSC 2000:: *49J45 Optimal control problems inv. semicontinuity and convergence
49J15 Optimal control problems with ODE (existence)
49J99 Existence theory for optimal solutions

Keywords: limit problem; $\Gamma$-convergence

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