Geodesic
On controllability of nonlinear distributed systems on a~set of discretized controls
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 1 (2013), pp. 83-98

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For nonlinear distributed systems representable as a Volterra functional operator equation in a Lebesgue space, sufficient conditions for pointwise controllability with respect to a nonlinear functional are proved. The controls are assumed to belong to a given set $\mathcal D$ of piecewise constant vector functions id est can be regarded as discretized controls. For the equation under study we define the set $\Omega$ of global solvability as the set of all admissible controls for which the equation has a global solution. As an auxiliary result having a separate interest, we also establish under our hypotheses the equality $\Omega=\mathcal D$. The reduction of controlled distributed systems to the functional operator equation under study is illustrated by two examples, namely a Dirichlet boundary value problem for a second order parabolic equation and a mixed boundary value problem for a second order hyperbolic equation; both equations of a rather general form.
Keywords: nonlinear distributed systems, controllability, discretized controls, Volterra functional operator equation. 
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     author = {A. V. Chernov},
     title = {On controllability of nonlinear distributed systems on a~set of discretized controls},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {83--98},
     publisher = {mathdoc},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2013_1_a7/}
}
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A. V. Chernov. On controllability of nonlinear distributed systems on a~set of discretized controls. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 1 (2013), pp. 83-98. http://geodesic.mathdoc.fr/item/VUU_2013_1_a7/