PDE control using viability and reachability analysis Alexandre Bayen Jean-Pierre Aubin Patrick Saint-Pierre Philadelphia, March 29 th, 2004

Known capture basins and viability kernels in everyday life [Mitchell, 2001]

Outline I.The capture basin and viability kernel II.The capture basin: an abstraction to solve a PDE I.Epigraphical solution II.Two canonical examples in path planning and optimal control III.The capture basin as an abstraction for a PDE control problem I.Controlling the PDE through its graph II.The set-valued viability solution III.A computational example IV.Towards a selection criterion for uniqueness Motivation: The capture basin is an efficient abstraction to use in order to solve PDE control problems.

Definition of a capture basin For a set valued dynamics One can define the set of trajectories Given a constraint set and a target The capture basin of under the constraint and the dynamics, denoted by is defined as the set of points of points of such that there exists at least one trajectory which reaches in a finite time and stays in for all [Aubin, 1991]

Illustration of the capture basin [Mitchell, Bayen, Tomlin, HSCC 2001] Constraint set: flight envelope Target set: set admissible touch down parameters Landing envelope: set of flight parameters from which a safe touch down is possible is the capture basin Landing envelope of a DC9-30 aircraft K C

Illustration of the capture basin K C C C K

Viability – invariance – reachability… Differential games: Isaacs, Basar, Lewin Reachability: Tomlin, Lygeros, Pappas, Sastry, Mitchell, Bayen, Kurzhanski, Varaiya, Maler, Krogh, Dang, Feron, Lynch Viability: Aubin, Saint-Pierre, Cardaliaguet, Quincampoix, Saint-Pierre, Cruck Viscosity solutions of HJE : Lions, Evans, Crandall, Frankowska, Bardi, Capuzzo- Dolcetta, Falcone, Branicky, Sethian, Vladimirsky Invariance: Sontag, Clarke, Leydaev, Stern, Wolenski, Khalil Optimal control, bisimulations: Broucke, Sangiovanni-Vincentelli, Di Benedetto Lyapunov theory, invariance basins: Sontag, Kokotovic, Krstic, Leitmann

Outline I.The capture basin and viability kernel II.The capture basin: an abstraction to solve a PDE I.Epigraphical solution II.Two canonical examples in path planning and optimal control III.The capture basin as an abstraction for a PDE control problem I.Controlling the PDE through its graph II.The set-valued viability solution III.A computational example IV.Towards a selection criterion for uniqueness

C How to compute the minimum time to reach C ? Example: One dimensional target C Set valued dynamics Add one dimension for time: Epigraph of the minimum time function

C Augment dynamics along the axis “count down” Dynamics along the horizontal axis Epigraph of the minimum time function

C Augment dynamics along the axis convex hullof the dynamicswith zero So that it is possible to stop in the target Epigraph of the minimum time function

C convex hullof the dynamicswith zero Epigraph of the minimum time function

C

C K [Cardaliaguet, Quincampoix, Saint-Pierre, 1997] Epigraph of the minimum time function

Outline I.The capture basin and viability kernel II.The capture basin: an abstraction to solve a PDE I.Epigraphical solution II.Two canonical examples in path planning and optimal control III.The capture basin as an abstraction for a PDE control problem I.Controlling the PDE through its graph II.The set-valued viability solution III.A computational example IV.Towards a selection criterion for uniqueness

optimal trajectory [Saint-Pierre, 2001] Example: minimum exit time Dynamics: in the domain on the target boundary on the domain boundary [Frankowska, 1994] [Bayen, Cruck, Tomlin, 2002] [Cardaliaguet, Quincampoix, Saint-Pierre, 1997]

Application to Air Traffic Control flying east at fixed heading. flying northwest at fixed heading. Available heading change (30 deg. west) any time When is the last time for to change heading so that is guaranteed to avoid collision ? [Bayen, Cruck, Tomlin, HSCC 2002]

Outline I.The capture basin and viability kernel II.The capture basin: an abstraction to solve a PDE I.Epigraphical solution II.Two canonical examples in path planning and optimal control III.The capture basin as an abstraction for a PDE control problem I.Controlling the PDE through its graph II.The set-valued viability solution III.A computational example IV.Towards a selection criterion for uniqueness

Characteristic system Consider the following characteristic system Consider a given function Extend it to 3 dimensions: Consider its graph as a target

Initial conditions only

Frankowska solution of the Burgers equation Theorem: viability solution is the unique Frankowska solution to the Burgers equation (1) satisfying the initial condition in the sense that

Application: the LWR equation General conservation law:

Application: the LWR equation Change the characteristic system: General conservation law: car density (normalized) car flux (cars / 5 min)

Outline I.The capture basin and viability kernel II.The capture basin: an abstraction to solve a PDE I.Epigraphical solution II.Two canonical examples in path planning and optimal control III.The capture basin as an abstraction for a PDE control problem I.Controlling the PDE through its graph II.The set-valued viability solution III.A computational example IV.Towards a selection criterion for uniqueness

Initial conditions only

Initial and boundary conditions

Initial and boundary conditions, constraints

Outline I.The capture basin and viability kernel II.The capture basin: an abstraction to solve a PDE I.Epigraphical solution II.Two canonical examples in path planning and optimal control III.The capture basin as an abstraction for a PDE control problem I.Controlling the PDE through its graph II.The set-valued viability solution III.A computational example IV.Towards a selection criterion for uniqueness

Computation with V IABILYS This computer: 3 years old, 800Mhz, 128 MRAM ©

Computational example [Oleinik, 1957], [Evans, 1998] [Aubin et al., 2004] [Ansorge 1995] Example: entropy solution

Viability solution Entropy solutionViability solution Jameson-Schmidt-Turkel Daganzo Lax-Friedrichs Analytical Analytical entropy solution Analytical viability solution Numerical viability solution

Outline I.The capture basin and viability kernel II.The capture basin: an abstraction to solve a PDE I.Epigraphical solution II.Two canonical examples in path planning and optimal control III.The capture basin as an abstraction for a PDE control problem I.Controlling the PDE through its graph II.The set-valued viability solution III.A computational example IV.Towards a selection criterion for uniqueness

Towards the selection of a unique selection General conservation law: Consider the cumulated [mass] of cars: Transformation into a Hamilton-Jacobi equation:

HJE with constraints Problem: control a Hamilton-Jacobi equation with constraints find a unique solution [selection] Construct a convex flux function: Theorem: The Baron-Jensen-Frankowska solution to the following Hamilton-Jacobi equation Is defined as the following capture basin: Where the characteristic system reads: with

Interpretation The Barron-Jensen-Frankowska is upper-semicontinuous “In practice”, it is continuous. It can be computed using the viability algorithm. Constraints can be incorporated into the solution (and the computation).

Summary The capture basin, initially defined in optimal control can be used as a good abstraction for solving a PDE. It can be used to control the graph of the solution of a PDE directly. Capture basins of dimension 3 can be computed very efficiently. The uniqueness problem can be resolved with a variable change through HJ equation. How to select the proper solution directly is an open problem.

[Aubin, Saint-Pierre, 2004] Discrete dynamical system Constant input u Initial condition x Which x are such that after an infinite number of iterations, is still in the ball Fractals: the Mandelbrot function

Fragility of the viability kernel [Aubin, Saint-Pierre, 2004]

Known capture basins and viability kernels in everyday life [Mitchell, 2001]

Initial and boundary conditions, constraints