Infinite dimensional systems´ sliding motions

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. We then compare this definition with an extended version of the equivalent control method for infinite dimensional systems proposed by V. Utkin and Yu. Orlov. We prove that, if the sliding manifold satisfies suitable regularity hypotheses, the projected evolution found by means of the equivalent control and our sliding mode do coincide. We then apply these results to the problem of stabilization of a heat equation.