Approximate controllability for nonlinear evolution hemivariational inequalities in Hilbert spaces. (English) Zbl 1330.47074
Let \(H\) be a separable Hilbert space with the scalar product \(\langle \cdot,\cdot\rangle_H\); \(\{A(t): t \in [0,b]\}\) a family of linear closed densely defined operators on \(H\) generating the evolution system \(\{U(t,s)\}\) of compact operators. The authors consider the following evolution hemivariational inequality
\[
\begin{cases} \langle-x^\prime (t) + A(t)x(t) + Bu(t),v\rangle_H + F^0(t,x(t);v) \geq 0 \quad \text{for all }t \in [0,b],\;v \in H; \cr x(0) = x_0. \end{cases}
\]
Here, \(B\) is a bounded linear operator from a Hilbert space \(H\) into \(H\), \(u(\cdot)\) is the control function and \(F^0(t,\cdot;\cdot)\) denotes the generalized directional derivative of a locally Lipschitz function \(F(t,\cdot) : H \to \mathbb{R}.\)
Some sufficient conditions are presented under which the approximate controllability of the associated linear system implies the approximate controllability of the above system. An example dealing with an initial-boundary value problem for a heat equation is given.
Some sufficient conditions are presented under which the approximate controllability of the associated linear system implies the approximate controllability of the above system. An example dealing with an initial-boundary value problem for a heat equation is given.
Reviewer: Valerii V. Obukhovskij (Voronezh)
MSC:
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
| 34G25 | Evolution inclusions |
| 49J52 | Nonsmooth analysis |
| 49J53 | Set-valued and variational analysis |
| 93B05 | Controllability |
| 93C20 | Control/observation systems governed by partial differential equations |
| 93C25 | Control/observation systems in abstract spaces |
Keywords:
approximate controllability; hemivariational inequality; differential inclusion; multivalued map; fixed pointReferences:
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