Using differential inclusions and viability theory we define sliding modes for (feedback) controlled semilinear differential equations in Banach spaces. We compare this definition with the equivalent control method for infinite-dimensional systems proposed by V. Utkin and Yu. Orlov. We show that if the sliding manifold satisfies suitable regularity hypotheses and the semigroup is compact, the projected evolution found by means of the equivalent control and our sliding mode do coincide.
"Sliding modes in Banach spaces." Differential Integral Equations 15 (2) 167 - 189, 2002.