| Abstract: |
| In a Hilbet space $H$ consider a semilinear control system $$X`(t)=AX(t)+f (t,X(t),u(t)), t>0, $$ where controls $u(\cdot)$ are Lebesgue measurable and take values in a prescribed set $U$. A closed subset $K \subset H$ is said to be invariant under this control system whenever all solutions starting from a point of $K$, at any time $t_0 \geq 0$, remain in $K$ as long as they exist. It is called viable whenever for any initial condition in $K$, at any time $t_0 \geq 0$, there exists a control $u(\cdot)$ such that the corresponding trajectory stays in $K$ forever. For a self-adjoint strictly dissipative operator $A$, perturbed by a (possibly unbounded) nonlinear term $f$, we give necessary and sufficient conditions for the invariance of $K$, formulated in terms of $A, \, f,$ and the distance function from $K$. Then, we also provide sufficient conditions for the viability of $K$ under the compactness assumption on the semigroup generated by $A$. Finally, we apply the above theory to a bilinear control problem for the heat equation in a bounded domain, where one is interested in keeping solutions in one fixed level set of a smooth integral functional. |
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