Optimal design via variational principles: The one-dimensional case. (English) Zbl 0985.49006

Bellido, José Carlos; Pedregal, Pablo

J. Math. Pures Appl., IX. Sér. 80, No. 2, 245-261 (2001).

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.
The optimal control problem considered is: find \((u,y)\) such that \[ u\in\{v\in L^\infty(0,1) : v(x)\in K\text{ for a.e. }x\in(0,1)\}, \]
\[ -(G(x,u(x),y(x),y'(x)))'=0\text{ in }(0,1),\;y(0)=y_0,\;y(1)=y_1, \]
\[ \int_0^1V(x,u(x),y(x),y'(x))dx\leq\gamma \] which minimizes the cost \[ I(u,y)=\int_0^1F(x,u(x),y(x),y'(x))dx, \] where \(K\subseteq{\mathbb R}^n\), \(G\), \(V\) and \(F\) are Carathéodory functions, and \(y_0\), \(y_1\) and \(\gamma\) are given.
Under suitable assumptions on the data, the authors prove an existence result for the above problem.
Several particular examples are also investigated where remarks on the non-existence are included.

Reviewer: Riccardo De Arcangelis (Napoli)

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
49J30	Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49Q20	Variational problems in a geometric measure-theoretic setting

Keywords:

optimal control; existence/nonexistence of solutions; convexity