## Domain invariance for local solutions of semilinear evolution equations in Hilbert spaces.(English)Zbl 1454.58010

Summary: A closed set $$K$$ of a Hilbert space $$H$$ is said to be invariant under the evolution equation $X^\prime ( t ) = A X ( t ) + f (t, X(t)) \quad ( t > 0 ),$ whenever all solutions starting from a point of $$K$$, at any time $$t_0 \geqslant 0$$, remain in $$K$$ as long as they exist.
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 give sufficient conditions for the viability of $$K$$ for the control system $X^\prime ( t ) = A X ( t ) + f (t, X(t), u(t)) \quad ( t > 0 , u ( t ) \in U ).$ Finally, we apply the above theory to a bilinear control problem for the heat equation in a bounded domain of $$\mathbb{R}^N$$, where one is interested in keeping solutions in one fixed level set of a smooth integral functional.

### MSC:

 58D25 Equations in function spaces; evolution equations 47H06 Nonlinear accretive operators, dissipative operators, etc. 37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems

### Keywords:

semilinear evolution equations in Hilbert spaces
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