Optimal control of an evolution problem involving time-dependent maximal monotone operators. (English) Zbl 1490.49006
Summary: We consider a control problem in a finite-dimensional setting, which consists in finding a minimizer for a standard functional defined by way of two continuous and bounded below functions and a convex function, where the control functions take values in a closed convex set and the state functions solve a differential system made up of a differential inclusion governed by a maximal monotone operator; and an ordinary differential equation with a Lipschitz mapping in the right-hand side. We first show the existence of a unique absolutely continuous solution of our system, by transforming it to a sole evolution differential inclusion, and then use a result from the literature. Secondly, we prove the existence of an optimal solution to our problem. The main novelties are: the presence of the time-dependent maximal monotone operators, which may depend as well as their domains on the time variable; and the discretization scheme for the approximation of the solution.
MSC:
| 49J21 | Existence theories for optimal control problems involving relations other than differential equations |
| 34H05 | Control problems involving ordinary differential equations |
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 34A60 | Ordinary differential inclusions |
Keywords:
absolutely continuous variation; maximal monotone operator; objective function; optimal solution; pseudo-distanceReferences:
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