Feedback Control for an Abstract Parabolic Equation.

Abstract

The abstract parabolic equation x = Ax (A sectionial) is marginally stable if the nullspace H_0 of A is non-trivial but e^(+A) is exponentially stable on a complement H_1. An example is u_t> = DELTA u with Neumann boundary conditions. Assume B has the form: Bx := -SIGMA_j,k*WEIRD GREEK LETTER_j,k*WEIRD GREEK LETTER_k(x) WEIRD GREEK LETTER_j and is such that y = EPSILON(QB)y(Q := projection on H_0 along H_1) is exponentially stable on H_0 for small EPSILON > 0. Then x = Ax + EPSILONBx is exponentially stable for 0 < EPSILON < EPSILON_0.