Abstract
We show how, using differential inclusions and viability theory it is
possible to define sliding modes for (feedback) controlled semilinear
differential equations in Banach spaces. In order to compare this
definition with the equivalent control method proposed by V. Utkin and
Yu. Orlov for infinite dimensional systems, we introduce the notion of
extended equivalent control. This allows the interpretation of the
sliding motion by a classical semigroup approach. Then we are able to
prove that, if the sliding manifold satisfies suitable regularity
hypotheses, the projected evolution found by means of the extended
equivalent control and our sliding mode do coincide. We then apply these
results to the problem of stabilization of a heat equation and of a
delay differential equation.
We show how, using differential inclusions and viability theory it is possible to define sliding modes for (feedback) controlled semilinear differential equations in Banach spaces. In order to compare this definition with the equivalent control method proposed by V. Utkin and Yu. Orlov for infinite dimensional systems, we introduce the notion of
extended equivalent control. This allows the interpretation of the sliding motion by a classical semigroup approach. Then we are able to prove that, if the sliding manifold satisfies suitable regularity hypotheses, the projected evolution found by means of the extended equivalent control and our sliding mode do coincide. We then apply these
results to the problem of stabilization of a heat equation and of a delay differential equation.