On continuity of solutions for parabolic control systems and input-to-state stability

Abstract

We study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions. This question naturally appears when working with boundary controlled, linear partial differential equations. Here, we focus on parabolic equations which allow for operator-theoretic methods such as the holomorphic functional calculus. Moreover, we investigate stronger conditions than continuity leading to input-to-state stability with respect to Orlicz spaces. This also implies that the notions of input-to-state stability and integral-input-to-state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite-dimensional.

19 pages, final version of preprint, Prop. 6 and Thm 7 have been generalised to arbitrary Banach spaces, the assumption of boundedness of the semigroup in Thm 10 could be dropped

References in corpus (3)

• The $H^{\infty }$-Functional Calculus and Square Function Estimates
• Strong input-to-state stability for infinite dimensional linear systems
• Input-to-State Stability with Respect to Boundary Disturbances for a Class of Semi-linear Parabolic Equations

Cited by in corpus (19)

• ISS Property with Respect to Boundary Disturbances for a Class of Riesz-Spectral Boundary Control Systems
• Exponential input-to-state stabilization of a class of diagonal boundary control systems with delay boundary control
• Noncoercive Lyapunov functions for input-to-state stability of infinite-dimensional systems
• Input-to-State Stability of a Clamped-Free Damped String in the Presence of Distributed and Boundary Disturbances
• Integral input-to-state stability of unbounded bilinear control systems
• A Desch-Schappacher perturbation theorem for bi-continuous semigroups
• Input-to-state stability for parabolic boundary control: Linear and semi-linear systems
• Well-posedness and Stability for Interconnection Structures of Port-Hamiltonian Type
• Boundary input-to-state stabilization of a damped Euler-Bernoulli beam in the presence of a state-delay
• Limit-case admissibility for positive infinite-dimensional systems
• A Note on the Maximum Principle-based Approach for ISS Analysis of Higher Dimensional Parabolic PDEs with Variable Coefficients
• Well-posedness and robust stability of a nonlinear ODE-PDE system
• On BIBO stability of infinite-dimensional linear state-space systems
• Admissible operators for sun-dual semigroups
• A refinement of Baillon's theorem on maximal regularity
• Approximations of Lyapunov functions for ISS analysis of a class of nonlinear parabolic PDEs
• A weak maximum principle-based approach for input-to-state stability analysis of nonlinear parabolic PDEs with boundary disturbances
• Input-to-State Stability of Nonlinear Parabolic PDEs with Dirichlet Boundary Disturbances
• Positive Desch-Schappacher perturbations of bi-continuous semigroups on $\mathrm{AM}$-spaces