Stabilization by Sparse Controls for a Class of Semilinear Parabolic Equations

Abstract

Abstract. Stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional. This formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state. Such a property is not present for commonly chosen control mechanisms. To establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons, which is otherwise not available.

A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying Parabolic Equations

“…Since we have now an estimate for γ 1 (·) in (56), it is of interest to study the effect of the constant δ on the constants α(δ, T ), c H (δ, T ), c V (δ, T ) given in Theorem 6.1 for a fixed T ≥ T * . Due to (18), as δ is getting smaller for δ < T 2 , α(δ, T ) becomes smaller. Moreover, by definitions of γ 1 (·) given in (56), we can infer that γ 1 (·) is an increasing function and it vanishes as δ → 0.…”

“…Remark 2.2. For fixed δ > 0, due to inequality (18) we have lim T →∞ α(T ) = 1. Thus RHC is asymptotically optimal.…”

Infinite Horizon Optimal Control Problems for a Class of Semilinear Parabolic Equations

“…Proof. To verify this result we can follow the proof of [7,Theorem 2.5]. For this purpose we use that our conditions on f imply that y(x, t)f (x, t, y(x, t)) ≥ 0 for almost every (x, t)…”

Semiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation

“…see pg. 521 of [7]. Given two Banach spaces X and Y , the space of linear and continuous functionals between X and Y is denoted by L(X, Y ).…”