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

Cannarsa, Piermarco; Da Prato, Giuseppe; Frankowska, Hélène

J. Lond. Math. Soc., II. Ser. 102, No. 1, 287-318 (2020).

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