Controllability of Semilinear Control Systems Dominated by the Linear Part

Controllability for Semilinear Functional Integrodifferential Equations

“…We consider the semilinear heat equation dealt with by Zhou [9], and Naito [4]. Let Then f is not uniformly bounded and R(g) ⊂ R(B) and from Theorem 3.2 it follows that the system of (SE1) is approximately controllable.…”

Controllability for Nonlinear Variational Evolution Inequalities

“…We formulate our nonlinear variational evolution inequality (NCE) as a semilinear control system in order to obtain the control problems. As in [7,9] we must assume the uniform boundedness of the nonlinear terms f (t, x) and (∂ϕ) 0 , where (∂ϕ) 0 : H → H is the minimum element of ∂ϕ. Since we apply the degree of mapping theorem in the proof of the main theorem, we need some compactness hypothesis.…”

“…The purpose of this paper is to show that the reachable set of the nonlinear variational inequality (SE) is equivalent to that of its corresponding linear variational inequality under the hypothesis (B) by applying results of [9] to the equation (NCE). We formulate our nonlinear variational evolution inequality (NCE) as a semilinear control system in order to obtain the control problems.…”

“…In [6], using more general hypotheses for nonlinear term f (·, x), we investigated the existence and the norm estimate of a solution of the above nonlinear equation on L 2 (0, T ; V ) ∩ W 1,2 (0, T ; V * ) considered as an equation in H as well as in V * , which is also applicable to optimal control problem. In the paper [9] Naito investigated the semilinear parabolic evolution equation in a Hilbert space H, in case where the nonlinear term f is a bounded uniformly continuous mapping from H to itself, and proved the approximate controllability under the following hypothesis;…”

Controllability of semilinear systems with fixed delay in control

“…is an orthonormal base for V and e n is the eigenfunction corresponding to the eigenvalue λ n = −n 2 of the operator A. Then the C 0 -semigroup T (t) generated by A has exp(λ n t) as the eigenvalues and e n as their corresponding eigenfunctions [12]. Define an infinite-dimensional spaceV bŷ…”