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This paper studies system theoretic properties of the class of difference inclusions of convex processes. We will develop a framework considering eigenvalues and eigenvectors, weakly and strongly invariant cones, and a decomposition of convex processes. This framework will lead to characterizations of reachability, stabilizability, and (null-)controllability for nonstrict convex processes in terms of spectral properties. These characterizations generalize all previously known results regarding, for instance, linear processes and specific classes of nonstrict convex processes.

Projected gradient descent denotes a class of iterative methods for solving optimization programs. In convex optimization, its computational complexity is relatively low whenever the projection onto the feasible set is relatively easy to compute. On the other hand, when the problem is nonconvex, e.g., because of nonlinear equality constraints, the projection becomes hard and thus impractical. In this paper, we propose a projected gradient method for nonlinear programs that only requires projections onto the linearization of the nonlinear constraints around the current iterate, similar to sequential quadratic programming (SQP). The proposed method falls neither into the class of projected gradient descent approaches, because the projection is not performed onto the original nonlinear manifold, nor into that of SQP, since second-order information is not used. For nonlinear smooth optimization problems, we assess local and global convergence to a Karush–Kuhn–Tucker point of the original problem. Further, we show that nonlinear model predictive control is a promising application of the proposed method, due to the sparsity of the resulting optimization problem., SIAM Journal on Control and Optimization, 56 (3), ISSN:0363-0129, ISSN:1095-7138

We apply methods of the so-called inverse problem of the calculus of variations to the stabilization of an equilibrium of a class of two-dimensional controlled mechanical systems. The class is general enough to include, among others, the inverted pendulum on a cart and the inertia wheel pendulum. By making use of a condition that follows from Douglas' classification, we derive feedback controls for which the control system is variational. We then use the energy of a suitable controlled Lagrangian to provide a stability criterion for the equilibrium.

We investigate necessary and sufficient conditions under which a general nonlinear affine control system with outputs can be written as a gradient control system corresponding to some pseudo-Riemannian metric defined on the state space. The results rely on a suitable notion of compatibility of the system with respect to a given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role in the discussion., Comment: 21 pages, no figures

In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are lines exactly when the induced norm on the horizontal space is strictly convex. As a further consequence, we show that all isometric embeddings between such homogeneous groups are affine. The core of the proof is an asymptotic study of the extremals given by the Pontryagin Maximum Principle., Comment: 19 pages, simplified Section 3 and proof of Theorem 1.1, extracted proof of Theorem 1.3 into an independent proposition. To appear in SIAM Journal on Control and Optimization

First Published in SIAM Journal on Control and Optimization in Volume 55, Issue 4, 2017, Pages 2437-2459, published by the Society for Industrial and Applied Mathematics (SIAM), This article is devoted to studying the null controllability of evolution equations with memory terms. The problem is challenging not only because the state equation contains memory terms but also because the classical controllability requirement at the final time has to be reinforced, involving the contribution of the memory term, to ensure that the solution reaches the equilibrium. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the adjoint system. We first consider finite-dimensional dynamical systems involving memory terms and derive rank conditions for controllability. Then the null controllability property is established for some parabolic equations with memory terms, by means of Carleman estimates, F. W. Chaves-Silva was partially supported by the ERC project Semi- Classical Analysis of Partial Di erential Equations, ERC-2012-ADG, project number: 320845. X. Zhang was supported by the NSF of China under grant 11231007 and the Chang Jiang Scholars Program from the Chinese Education Ministry. This work was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, ICON of the French ANR (ANR-2016-ACHN-0014-01), FA9550-15-1-0027 of AFOSR, A9550-14-1-0214 of the EOARD-AFOSR, and the MTM2014-52347 Grant of the MINECO (Spain)

The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of the turnpike is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike in the nondetectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems, we establish existence and convergence of solutions to a generalized differential Riccati equation., Comment: 31 pages, 1 figure

International audience; Self-organization and control around flocks and mills is studied for second-order swarming systems involving self-propulsion and potential terms. It is shown that through the action of constrained control, is it possible to control any initial configuration to a flock or a mill. The proof builds on an appropriate combination of several arguments: LaSalle invariance principle and Lyapunov-like decreasing functionals, control linearization techniques, and quasi-static deformations. A stability analysis of the second-order system guides the design of feedback laws for the stabilization to flock and mills, which are also assessed computationally.