The paper is divided in two parts. In the first part a deep investigation is made on some system theoretical aspects of periodic systems and control, including the notions of $H_2$ and $H_{\infty }$ norms, the parametrization of stabilizing controllers, and the existence of periodic solutions to Riccati differential equations and/or inequalities. All these aspects are useful in the second part, where some parametrization and control problems in $H_2$ and $H_{\infty }$ are introduced and solved.

This paper deals with some state-feedback $H_2/H_{\infty }$ control problems for continuous time periodic systems. The derivation of the theoretical results underlying such problems has been presented in the first part of the paper. Here, the parametrization and optimization problems in $H_2$, $H_{\infty }$ and mixed $H_2/H_{\infty }$ are introduced and solved.

Recent years have witnessed an increasing interest in coordinated control of distributed dynamic systems. In order to steer a distributed dynamic system to a desired state, it often becomes necessary to have a prior control over the graph which represents the coupling among interacting agents. In this paper, a simple but compelling model of distributed dynamical systems operating over a dynamic graph is considered. The structure of the graph is assumed to be relied on the underling system's states....

We seek to classify the full-rank left-invariant control affine systems evolving on solvable three-dimensional Lie groups. In this paper we consider only the cases corresponding to the solvable Lie algebras of types II, IV, and V in the Bianchi-Behr classification.

A modification of digital controller algorithms, based on the introduction of a virtual reference value, which never exceeds active constraints in the actuator output is presented and investigated for some algorithms used in single-loop control systems. This idea, derived from virtual modification of a control error, can be used in digital control systems subjected to both magnitude and rate constraints. The modification is introduced in the form of on-line adaptation to the control task. Hence...

For the Schrödinger equation, $(i{\partial }_t+\nabla )u=0$ on a torus, an arbitrary non-empty open set $\Omega$ provides control and observability of the solution: $∥u{\left|{}_{t=0}\right.∥}_{L^2\left({𝕋}^2\right)}\le K_T{∥u∥}_{L^2([0,T]\times \Omega )}$. We show that the same result remains true for $(i{\partial }_t+\nabla -V)u=0$ where $V\in L^2\left({𝕋}^2\right)$, and ${𝕋}^2$ is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for $V\in C\left({𝕋}^2\right)$ and conjectured for $V\in L^{\infty }\left({𝕋}^2\right)$. The higher dimensional generalization remains open for $V\in L^{\infty }\left({𝕋}^n\right)$.

Four optimal design problems and a weight minimization problem are considered for elastic plates with small bending rigidity, resting on a unilateral elastic foundation, with inner rigid obstacles and a friction condition on a part of the boundary. The state problem is represented by a variational inequality and the design variables influence both the coefficients and the set of admissible state functions. If some input data are allowed to be uncertain a new method of reliable solutions is employed....

For a general time-varying system, we prove that existence of an “Output Robust Control Lyapunov Function” implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well known result due to Coron and Rosier [J. Math. Syst. Estim. Control 4 (1994) 67–84] concerning stabilization of autonomous systems by means of time-varying periodic...

For a general time-varying system, we prove that existence of an “Output Robust Control Lyapunov Function” implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well known result due to Coron and Rosier [J. Math. Syst. Estim. Control4 (1994) 67–84] concerning stabilization of autonomous systems by means of time-varying...

This paper presents a method to design explicit control Lyapunov functions for affine and homogeneous systems that satisfy the so-called “Jurdjevic-Quinn conditions”. For these systems a positive definite function V0 is known that can only be made non increasing by feedback. We describe how a control Lyapunov function can be obtained via a deformation of this “weak” Lyapunov function. Some examples are presented, and the linear quadratic situation is treated as an illustration.

A control system of the second order in time with control $u=u\left(t\right)\in L^2([0,T];U)$ is considered. If the system is controllable in a strong sense and uT is the control steering the system to the rest at time T, then the L2–norm of uT decreases as $1/\sqrt{T}$ while the $L^1([0,T];U)$–norm of uT is approximately constant. The proof is based on the moment approach and properties of the relevant exponential family. Results are applied to the wave equation with boundary or distributed controls.

We consider a controllability problem for a beam, clamped at one boundary and free at the other boundary, with an attached piezoelectric actuator. By Hilbert Uniqueness Method (HUM) and new results on diophantine approximations, we prove that the space of exactly initial controllable data depends on the location of the actuator. We also illustrate these results with numerical simulations.

A delay stochastic method is introduced to control a certain class of chaotic systems. With the Lyapunov method, a suitable kind of controllers with multiplicative noise is designed to stabilize the chaotic state to the equilibrium point. The method is simple and can be put into practice. Numerical simulations are provided to illustrate the effectiveness of the proposed controllable conditions.