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A uniformly controllable and implicit scheme for the 1-D wave equation

Arnaud Münch (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δ t . We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h 2 and Δ t 2 . Using a discrete...

A uniformly controllable and implicit scheme for the 1-D wave equation

Arnaud Münch (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper studies the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. Using...

A weak regularity theorem for real analytic optimal control problems.

Hector J. Sussmann (1986)

Revista Matemática Iberoamericana

We consider real analytic finite-dimensional control problems with a scalar input that enters linearly in the evolution equations. We prove that, whenever it is possible to steer a state x to another state y by means of a measurable control, then it is possible to steer x to y by means of a control that has an extra regularity property, namely, that of being analytic on an open dense subset of its interval of definition. Since open dense sets can have very small measure, this is a very weak property....

Alternate checking criteria for reachable controllability of rectangular descriptor systems

Vikas Kumar Mishra, Nutan Kumar Tomar (2017)

Kybernetika

Contrary to state space systems, there are different notions of controllability for linear time invariant descriptor systems due to the non smooth inputs and inconsistent initial conditions. A comprehensive study of different notions of controllability for linear descriptor systems is performed. Also, it is proved that reachable controllability for general linear time invariant descriptor system is equivalent to the controllability of some matrix pair under an assumption milder than impulse controllability....

An approach to the analysis of observability and controllability in nonlinear systems via linear methods

Alexey Zhirabok, Alexey Shumsky (2012)

International Journal of Applied Mathematics and Computer Science

The paper is devoted to the problem of observability and controllability analysis in nonlinear dynamic systems. Both continuous- and discrete-time systems described by nonlinear differential or difference equations, respectively, are considered. A new approach is developed to solve this problem whose features include (i) consideration of systems with non-differentiable nonlinearities and (ii) the use of relatively simple linear methods which may be supported by existing programming systems, e.g.,...

An estimation of the controllability time for single-input systems on compact Lie Groups

Andrei Agrachev, Thomas Chambrion (2006)

ESAIM: Control, Optimisation and Calculus of Variations

Geometric control theory and Riemannian techniques are used to describe the reachable set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to estimate the minimal time needed to reach any point from identity. This method provides an effective way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum system with a drift from a given initial position to a given final position. The bounds include diameters...

An optimal strong equilibrium solution for cooperative multi-leader-follower Stackelberg Markov chains games

Kristal K. Trejo, Julio B. Clempner, Alexander S. Poznyak (2016)

Kybernetika

This paper presents a novel approach for computing the strong Stackelberg/Nash equilibrium for Markov chains games. For solving the cooperative n -leaders and m -followers Markov game we consider the minimization of the L p - norm that reduces the distance to the utopian point in the Euclidian space. Then, we reduce the optimization problem to find a Pareto optimal solution. We employ a bi-level programming method implemented by the extraproximal optimization approach for computing the strong L p - Stackelberg/Nash...

An output controllability problem for semilinear distributed hyperbolic systems

E. Zerrik, R. Larhrissi, H. Bourray (2007)

International Journal of Applied Mathematics and Computer Science

The paper aims at extending the notion of regional controllability developed for linear systems cite to the semilinear hyperbolic case. We begin with an asymptotically linear system and the approach is based on an extension of the Hilbert uniqueness method and Schauder's fixed point theorem. The analytical case is then tackled using generalized inverse techniques and converted to a fixed point problem leading to an algorithm which is successfully implemented numerically and illustrated with examples....

Analyse régionale des systèmes distribués

A. El Jai (2002)

ESAIM: Control, Optimisation and Calculus of Variations

Le but de cet article est de montrer l’état de l’art dans l’analyse des systèmes distribués lorsque l’on examine certains concepts à partir de considérations régionales. Autrement dit, à partir de la donnée d’un système dynamique défini sur un domaine Ω , on ne s’intéresse à sa contrôlabilité, à son observabilité, à sa stabilité, ... que sur une région privilégiée ω , ω ⊂ Ω . Partant de concepts classiques, on développe leur adaptation au cas régional. On développe ensuite des concepts régionaux propres...

Analyse régionale des systèmes distribués

A. El Jai (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Le but de cet article est de montrer l'état de l'art dans l'analyse des systèmes distribués lorsque l'on examine certains concepts à partir de considérations régionales. Autrement dit, à partir de la donnée d'un système dynamique défini sur un domaine Ω, on ne s'intéresse à sa contrôlabilité, à son observabilité, à sa stabilité, ... que sur une région privilégiée ω, ω ⊂ Ω . Partant de concepts classiques, on développe leur adaptation au cas régional. On développe ensuite des concepts régionaux...

Analysis of some dual properties in discrete dynamic systems

Aleksey Zhirabok (2006)

International Journal of Applied Mathematics and Computer Science

The problem of duality in nonlinear and linear systems is considered. In addition to the known duality between controllability and observability, new dual notions and their properties are investigated. A way to refine these properties through an isomorphic transformation of the original systems is suggested.

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