Abstract
In this article, we deal with the existence, uniqueness, and a variation of solutions of the nonlinear control system with nonlinear monotone hemicontinuous and coercive operator. Moreover, the approximate controllability for the given nonlinear control system is studied.
Highlights
Let H and V be two real separable Hilbert spaces such that V is a dense subspace of H.We are interested in the following nonlinear differential control system on H: ⎧⎪⎨ x (t) + Ax(t) = g(t, xt, t k(t, s, xs)ds) + (Bu)(t),⎪⎩ x(0) = φ0, x(s) = φ1(s) − h ≤ s ≤ 0, 0 < t, (SE)where the nonlinear term, which is a Lipschitz continuous operator, is a semilinear version of the quasi-linear form
If the right-hand side of the equation (SE) belongs to L2(0, T; V* ), it is well known as the quasi-autonomous differential equation(see Theorem 2.6 of Chapter III in [1])
The problem of existence for solutions of semilinear evolution equations in Banach spaces has been established by several authors [1,2,3]
Summary
Let H and V be two real separable Hilbert spaces such that V is a dense subspace of H. The controller B is a linear-bounded operator from a Banach space L2(0, T; U) to L2(0, T; H) for any T >0. The problem of existence for solutions of semilinear evolution equations in Banach spaces has been established by several authors [1,2,3]. The previous results on the approximate controllability of a semilinear control system have been proved as a particular case of sufficient conditions for the approximate solvability of semilinear equations, assuming either that the semigroup generated by A is a compact operator or that the corresponding linear system (SE) when g ≡ 0 is approximately controllable. Triggian [12] proved that the abstract linear system is never exactly controllable in an infinite dimensional space when the semigroup generated by A is compact.
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