On controllability for a nonlinear Volterra equation. (English) Zbl 0768.93011

Summary: We consider the following nonlinear Volterra wave equation with a control function \(h(t)\): \[ u_{tt}=u_{xx}-\int^ t_ 0 k(t,s)f(u(s,x))dx+b(x)h(t),\quad 0<t<T,\quad 0<x<\ell\tag{1} \] with the boundary condition \(u(t,0)=u(t,\ell)=0\), \(0<t<T\) and with the initial condition \(u(0,x)=u_ t(0,x)=0\), \(0\leq x\leq\ell\). We prove approximate controllability of nonlinear systems (1) without any local restriction on reachable sets by showing the relation between the reachable sets of (1) and those of its corresponding linear system.


93B05 Controllability
93C10 Nonlinear systems in control theory
Full Text: DOI


[1] Bergh, J.; Löfstrom, J., Interpolation Spaces, An Introduction (1976), Springer: Springer Berlin · Zbl 0344.46071
[2] Cirina, M., Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control Optim., 7, 198-212 (1969) · Zbl 0182.20203
[3] Fattorini, H. O., Local controllability of a nonlinear wave equation, Math. Systems Theory, 9, 30-45 (1975) · Zbl 0319.93009
[4] Naito, K., Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim., 25, 715-722 (1987) · Zbl 0617.93004
[5] Naito, K.; Seidman, T. I., Invariance of the approximately reachable set under nonlinear perturbations, SIAM J. Control Optim., 29, 731-750 (1991) · Zbl 0729.49022
[6] Narukawa, K., Admissible controllability of vibrating systems with constrained control, SIAM J. Control Optim., 20, 770-783 (1982) · Zbl 0511.93042
[7] Narukawa, K., Complete controllability of one-dimensional vibrating systems with bang-bang controls, SIAM J. Control Optim., 22, 788-804 (1984) · Zbl 0568.49023
[8] Narukawa, K.; Suzuki, T., Nonharmonic Fourier series and its applications, Appl. math. Optim., 14, 249-264 (1986) · Zbl 0654.42016
[9] Pazy, A., Semigroups of linear operators and its applications to partial differential equations, (Appl. Math. Sci., 44 (1983), Springer: Springer Berlin) · Zbl 0516.47023
[10] Russel, D. L., Nonharmonic Fourier series in the control theory of distributed parameter systems, J. math. Analysis Applic., 18, 542-560 (1967) · Zbl 0158.10201
[11] Seidman, T. I., Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim., 25, 1173-1191 (1987) · Zbl 0626.49018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.