Domain invariance for local solutions of semilinear evolution equations in Hilbert spaces

A closed set K of a Hilbert space H is said to be invariant under the evolution equationX ′ ( t ) = A X ( t ) + f t , X ( t )( t > 0 ) , whenever all solutions starting from a point of K , at any timet 0 ⩾ 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 systemX ′ ( t ) = A X ( t ) + f t , X ( t ) , u ( t )( t > 0 , u ( t ) ∈ U ) . Finally, we apply the above theory to a bilinear control problem for the heat equation in a bounded domain of R N , where one is interested in keeping solutions in one fixed level set of a smooth integral functional.