Optimal design via variational principles: The one-dimensional case (Q5932263)

In the one-dimensional case, the authors study some optimal control problems like optimal design or structural optimization problems. In order to avoid the non-local character of the state equation, they recast the problems in a purely variational format, and explore how far one can go in proving existence results or analyzing non-existence situations.NEWLINENEWLINENEWLINEThe optimal control problem considered is: find \((u,y)\) such that NEWLINE\[NEWLINEu\in\{v\in L^\infty(0,1) : v(x)\in K\text{ for a.e. }x\in(0,1)\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE-(G(x,u(x),y(x),y'(x)))'=0\text{ in }(0,1),\;y(0)=y_0,\;y(1)=y_1,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\int_0^1V(x,u(x),y(x),y'(x))dx\leq\gammaNEWLINE\]NEWLINE which minimizes the cost NEWLINE\[NEWLINEI(u,y)=\int_0^1F(x,u(x),y(x),y'(x))dx,NEWLINE\]NEWLINE where \(K\subseteq{\mathbb R}^n\), \(G\), \(V\) and \(F\) are Carathéodory functions, and \(y_0\), \(y_1\) and \(\gamma\) are given.NEWLINENEWLINENEWLINEUnder suitable assumptions on the data, the authors prove an existence result for the above problem.NEWLINENEWLINENEWLINESeveral particular examples are also investigated where remarks on the non-existence are included.