Patrick Saint-Pierre

Set-Valued Numerical Analysis for Optimal Control and Differential Games

This chapter deals with theoretical and numerical results for solving qualitative and quantitative control and differential game problems. These questions are treated in the framework of set-valued analysis and viability theory. In a way, this approach is rather well adapted to look at these several problems with a unified point of view. The idea is to characterize the value function as a viability kernel instead of solving a Hamilton—Jacobi—Bellmann equation. This allows us to easily take into account state constraints without any controllability assumptions on the dynamic, neither at the boundary of targets, nor at the boundary of the constraint set. In the case of two-player differential games, the value function is characterized as a discriminating kernel. This allows dealing with a large class of systems with minimal regularity and convexity assumptions. Rigorous proofs of the convergence, including irregular cases, and completely explicit algorithms are provided.

Dynamic Management of Portfolios with Transaction Costs under Tychastic Uncertainty

We use in this chapter the viability/capturability approach for studying the problem of dynamic valuation and management of a portfolio with transaction costs in the framework of tychastic control systems (or dynamical games against nature) instead of stochastic control systems. Indeed, the very definition of the guaranteed valuation set can be formulated directly in terms of guaranteed viable-capture basin of a dynamical game.

Computing Viable Sets and Reachable Sets to Design Feedback Linearizing Control Laws Under Saturation

We consider feedback linearizable systems subject to bounded control input and nonlinear state constraints. In a single computation, we synthesize 1) parameterized nonlinear controllers based on feedback linearization, and 2) the set of states over which this controller is valid. This is accomplished through a reachability calculation, in which the state is extended to incorporate input parameters. While we use a Hamilton-Jacobi formulation, a viability approach is also feasible. The result provides a mathematical guarantee that for all states within the computed set, there exists a control law that simultaneously satisfy two separate goals: envelope protection (no violation of state constraints), and stabilization despite saturation. We apply this technique to two real-world systems: the longitudinal dynamics of a civil jet aircraft, and a two-aircraft, planar collision avoidance scenario. The result, in both cases, is a feasible range of input parameters for the nonlinear control law, and a corresponding controlled invariant set

Hybrid Kernels and Capture Basins for Impulse Constrained Systems

We investigate, for constrained controlled systems with impulse, the subset of initial positions contained in a set K from which starts at least one run viable in K - the hybrid viability kernel - eventually until it reaches a given closed target in finite time - the hybrid capture basin. We define a constructive algorithm which approximates this set. The knowledge of this set is essential for control problem since it provides viable hybrid feed-backs and viable runs. We apply this method for approximatingt he Minimal Time-to-reach Function in the presence of both constraints and impulses. Two examples are presented, the first deals with a dynamical system revealingthe complexity of the structure of hybrid kernels, the second deals with a Minimal Time problem with impulses.

Differential Games Through Viability Theory: Old and Recent Results

This article is devoted to a survey of results for differential games obtained through Viability Theory. We recall the basic theory for differential games (obtained in the 1990s), but we also give an overview of recent advances in the following areas: games with hard constraints, stochastic differential games, and hybrid differential games.We also discuss several applications.

Population viability in three trophic-level food chains

The perpetuation of three-trophic level ecosystems where the three species exhibit unpredictable time-varying survival strategies is described by a specific set, the viability kernel, gathering all states from which there exists at least one trajectory safeguarding each species over a given density threshold. The strategies permitting this property are delineated and called viable strategies. All solutions starting outside the viability kernel lead to too low densities or extinction. The viability approach highlights the timing of strategy changes necessary for a system to perpetuate itself or alternatively to lead one species to extinction. The study of the dependence of the viability kernel on the admissible sets of strategies reveals the minimal flexibilities necessary for the existence of the system. The shape of the viability kernel determines whether the exogenous addition or substraction of prey or predator will endanger the system or not, thus gathering different experiments with opposite results. The comparison of the coexistence kernel with viability kernels for one, two or three species points out the importance of repeated strategies, not necessarily in a periodic manner, thus emphasizing the concept of repetitions in ecosystems instead of cycles as a key feature of coexistence.

Approximation of slow solutions to differential inclusions

To approach a viable solution of a differential inclusion, i.e., staying at any time in a closed convexK, a sufficient condition is given implying the convergence of an approximation sequence defined from the Euler or Runge-Kutta methods applied to a selection process which corresponds to the slowsolution concept. WhenK is smooth, the convergence condition is satisfied. This proves that the method is implementable on a computer for solving, for instance, differentiable equations with a noncontinuous right-hand side. Since the usual best approximation operator is difficult to implement, we introduce a class of quasi-projectors much more suitable for computation.

The Hybrid Guaranteed Capture Basin Algorithm in Economics

Reaching a target while remaining in a given set for impulse dynamics can be characterized by a non deterministic controlled differential equation and a controlled instantaneous reset equation. The set of initial conditions from which a given objective can be reached is calculated using the Hybrid Guaranteed Capture Basin Algorithm. This algorithm was developed in finance to evaluate options in absence of impulse but in the presence of uncertainty and in control to evaluate minimal time functions to reach a target, in the presence of impulse but in absence of uncertainty. We study the problem of reaching a target in the presence of both impulse and uncertainty and present two applications in economics.

A Tychastic Approach to Guaranteed Pricing and Management of Portfolios under Transaction Constraints

Dynamic guaranteed pricing and management of a portfolio under transaction constraints is actually a problem straightforwardly set in terms of guaranteed capture basin of a time-dependent target that is viable in a time-dependent environment under (stochastic or tychastic) uncertain systems. The knowledge of the properties of “capture basin” of targets viable in evolving environments under an uncertain evolutionary system can be used for obtaining the corresponding properties for portfolios. They yield at each time both the evaluation of the capital and the transaction rule. They can be computed by viability algorithms and software providing the valuation of optimal portfolio and the management of their evolution. The capital function, which is actually the value function of a differential game, is the solution to a free boundary problem for nonlinear partial differential equations with discontinuous coefficients. This survey provides several examples.

Viability Kernels and Capture Basins for Analyzing the Dynamic Behavior: Lorenz Attractors, Julia Sets, and Hutchinson’s Maps

Some issues on chaotic solutions, to Lorenz systems, for instance, are related to the concepts of viability kernels of subsets under continuous time systems, or in the case of Julia or Cantor sets, for instance, under discrete time systems.