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In this paper, we study the null controllability of a coupled parabolic–hyperbolic system in one dimension with a single control using the moment method. More precisely, we consider a system coupling Kuramoto–Sivashinsky–Korteweg–de Vries equation and transport equation through first-order derivatives. We explore the null controllability of four different control systems with the control acting either on the periodic boundary or in some open subset of the interior of the domain with periodic boundary conditions. Depending on the position of the control, we get some regular periodic Sobolev space as the space of initial data for which the null controllability holds, provided the time is sufficiently large. [ABSTRACT FROM AUTHOR]

We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from H α (R , U) to H β (R , U) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail. [ABSTRACT FROM AUTHOR]

We study integral input-to-state stability of bilinear systems with unbounded control operators and derive natural sufficient conditions. The results are applied to a bilinearly controlled Fokker–Planck equation. [ABSTRACT FROM AUTHOR]

In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval [ 0 , T ] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals [ 0 , T ] give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form [ t - t / 2 n , t + (T - t) / 2 n ] is of the order 1 / min { t n , (T - t) n } . We also show that a similar result holds for ϵ -optimal solutions of the optimal control problems if ϵ > 0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied. [ABSTRACT FROM AUTHOR]

Linear differential systems, in Willems' behavioral system theory, are defined to be the solution sets to systems of linear constant coefficient PDEs, and they are naturally parameterized in a bijective way by means of polynomial modules. In this article, introducing appropriate topologies, this parametrization is made continuous in both directions. Moreover, the space of linear differential systems with a given complexity polynomial is embedded into a Grassmannian. [ABSTRACT FROM AUTHOR]

We consider forced Lur'e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay and partial differential equations are known to belong to this class of infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of the internal control of a parabolic PDE with nonlinear diffusion, nonlocal in space. In our main result, we prove the local exact controllability to the trajectories with distributed controls, locally supported in space. The main ingredients of the proof are a compactness–uniqueness argument and Kakutani's fixed-point theorem in a suitable functional setting. Some possible extensions and open problems concerning other nonlocal systems are presented. [ABSTRACT FROM AUTHOR]

This paper deals with a non-standard infinite dimensional linear quadratic control problem arising in the physics of non-stationary states (see, for example, Bertini et al. J Statist Phys 116:831-841, 2004): finding the minimum energy to drive a fixed stationary state $$\bar{x}=0$$ into an arbitrary non-stationary state x. The Riccati equation (RE) associated with this problem is not standard since the sign of the linear part is opposite to the usual one, thus preventing the use of the known theory. Here we consider the finite horizon case when the leading semigroup is exponentially stable. We prove that the linear selfadjoint operator P( t), associated with the value function, solves the above-mentioned RE (Theorem 4.12). Uniqueness does not hold in general, but we are able to prove a partial uniqueness result in the class of invertible operators (Theorem 4.13). In the special case where the involved operators commute, a more detailed analysis of the set of solutions is given (Theorems 4.14, 4.15 and 4.16). Examples of applications are given. [ABSTRACT FROM AUTHOR]