Viability Kernel Research Articles

Aircraft Take-Off in Windshear: A Viability Approach

This paper is devoted to the analysis of aircraft dynamics during tage-off in the presence of windshear. We formulate the take-off problem as a differential game against Nature. Here, the first player is the relative angle of attack of the aircraft (considered as the control variable) and the second player is the disturbance caused by a windshear. We impose state constraints on the state variables of the game, which represents aircraft safety constraints (minimum altitude, given altitude rate). By using viability theory, we address the question of existence of an open loop control assuring a viable trajectory (i.e. satisfying the state constraints) no matter the disturbance is, i.e. for all admissible disturbances causeed by the windshear. Through numerical simulations of the viability kernel algorithm, we demonstrate the capabilities of this approach for determining safe flight domains of an aircraft during take-off within windshear.

Numerical Schemes for Discontinuous Value Functions of Optimal Control

In this paper we explain that various (possibly discontinuous) value functions for optimal control problem under state-constraints can be approached by a sequence of value functions for suitable discretized systems. The key-point of this approach is the characterization of epigraphs of the value functions as suitable viability kernels. We provide new results for estimation of the convergence rate of numerical schemes and discuss conditions for the convergence of discrete optimal controls to the optimal control for the initial problem.

Open-Loop Viable Control Under Uncertain Initial State Information

We consider the problem of open-loop viable control of a nonlinear system in Rn in the case of a nonexactly known initial state. We characterize the family of those initial sets for which the problem is solvable. The characterization employs the notion of a contingent field to a given collection of sets introduced in the paper. It also involves an appropriate set-dynamic equation that describes the evolution of the state estimation within a prescribed collection of sets. An extension of the classical concept of viability kernel with respect to this set-dynamic equation is the key tool. We present an approximation scheme for the viability kernel which is numerically realizable in the case of low dimension and simple collections of sets chosen for state estimation (balls, ellipsoids, polyhedrons, etc.). As an application, we consider a viability differential game, where the uncertainty may enter also in the dynamics of the system as an input which is not known in advance. The control is then sought as a nonanticipative strategy depending on the uncertain input.

On the viability of a physiological system. regulation of potassium by aldosterone

Physiological systems maintain their variables within limits through regulatory processes. We have studied potassium regulation on the basis of the viability theory. The viability kernel is determined by taking into account the constraints of the variables of the dynamical system, i.e. the set of initial states for which there exists at least one viable trajectory. The set of regulations which maintain the viability of the physiological system is also determined. Finally, we investigate the influence of various strategies that allow the choice of one control among the viable regulations.

Multi-Target Control Problems

We study control problems with several targets in the case of nonlinear dynamic systems. The map associating with every initial condition the minimal time to reach successively two given targets is characterized in the framework of differential inclusions through the notion of viability kernel. This approach allows one to treat the problem without assumptions of regularity and to build numerical schemes computing the minimal time. We also study the problem where an order of visit of the targets is required. The statements are also extended to the case of p targets under state constraints. Equivalent formulations in terms of Hamilton–Jacobi equations are also provided.

Scale of Viability and Minimal Time of Crisis

In this paper, we introduce and study the minimal time of a crisis map which measures the minimal time spent outside a given closed domain of constraints by trajectory solutions of a differential inclusion. The interest of such a notion is basically to tackle simultaneously viability and target issues. The main mathematical result characterizes the epigraph of the crisis map in terms of a viability kernel of an augmented problem. This allows the obtaining of the numerical schemes we specify and to derive an equivalent Hamilton–Jacobi formulation. A simple economic example illustrates the results.

The Viability Kernel Algorithm for Computing Value Functions of Infinite Horizon Optimal Control Problems

We characterize in this paper the epigraph of the value function of a discounted infinite horizon optimal control problem as the viability kernel of an auxiliary differential inclusion. Then the viability kernel algorithm applied to this problem provides the value function of the discretized optimal control problem as the supremum of a nondecreasing sequence of functions iteratively defined. We also use the fact that an upper Painlevé–Kuratowski limit of closed viability domains is a viability domain to prove the convergence of the discrete value functions.

Lattice operators underlying dynamic systems

This paper investigates algebraic and continuity properties of increasing set operators underlying dynamic systems. We recall algebraic properties of increasing operators on complete lattices and some topologies used for the study of continuity properties of lattice operators. We apply these notions to several operators induced by a differential equation or differential inclusion. We especially focus on the operators associating with any closed subset its reachable set, its exit tube, its viability kernel or its invariance kernel. Finally, we show that morphological operators used in image processing are particular cases of operators induced by constant differential inclusion.

Approximation of the viability kernel

We study recursive inclusionsxn+1ź G(xn). For instance, such systems appear for discrete finite-difference inclusionsxn+1 źGź(xn) whereGź:=1+źF. The discrete viability kernel ofGź, i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withxn+1 źźź(xn) whereźź(x) =x + źF(x) + (ML/2) ź2ź. Secondly, we show that it can be approached by finite viability kernels associated withxhn+1 ź (źź(xhn+1) +ź(hź) źXh.

Viability of boundary of the viability kernel

Viability of boundary of the viability kernel

A Survey of Viability Theory

Some theorems of viability theory which are relevant to nonlinear control problems with state constraints and state-dependent control constraints are motivated and surveyed. They all deal with viable solutions to nonlinear control problems, i.e., solutions satisfying at each instant given state constraints of a general and diverse nature. Some classical results on controlled invariance of smooth nonlinear systems are adopted to the nonsmooth case, including inequality constraints bearing on the state and state-dependent constraints on the controls. For instance, existence of a viability kernel of a closed set (corresponding to the largest controlled invariant manifold) is provided under general conditions, even when the zero-dynamics algorithm does not converge. The concepts of slow and heavy viable solutions are introduced, providing concrete ways of regulating viable solutions, by closed-loop feedbacks and closed-loop dynamical feedbacks. Viability theorems also allow the extension of Lyapunov’s second method to nonsmooth observation functions and the construction of “best” Lyapunov functions. As an application, “fuzzy differential inclusion” is presented. Proofs and complements can be found in [Viability Theory, to appear, 1991]. They rely on properties of differential inclusion (see [Differential Inclusions, Springer-Verlag, Berlin, New York, 1984]) and set-valued analysis, (see [Set-Valued Analysis, Birkhäuser, Basel, 1990]).

Applied Functional Analysis

The Projection Theorem. Theorems on Extension and Separation. Dual Spaces and Transposed Operators. The Banach Theorem and the Banach-Steinhaus Theorem. Construction of Hilbert Spaces. L ~ 2 Spaces and Convolution Operators. Sobolev Spaces of Functions of One Variable. Some Approximation Procedures in Spaces of Functions. Sobolev Spaces of Functions of Several Variables and the Fourier Transform. Introduction to Set-Valued Analysis and Convex Analysis. Elementary Spectral Theory. Hilbert-Schmidt Operators and Tensor Products. Boundary Value Problems. Differential-Operational Equations and Semigroups of Operators. Viability Kernels and Capture Basins. First-Order Partial Differential Equations. Selection of Results. Exercises. Bibliography. Index.