PDE control using viability andreachability analysis

Alexandre Bayen Jean-Pierre Aubin Patrick Saint-Pierre

Philadelphia, March 29th, 2004

Known capture basins and viability kernelsin everyday life

[Mitchell, 2001]

Outline

I. The capture basin and viability kernelII. The capture basin: an abstraction to solve a PDE

I. Epigraphical solutionII. 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 graphII. The set-valued viability solutionIII. 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

CCK

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 PDEI. Epigraphical solutionII. 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 graphII. The set-valued viability solutionIII. A computational example

IV. Towards a selection criterion for uniqueness

C

How to compute the minimum time to reach C ?

Example: One dimensional target CSet 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 hull of the dynamics with zero

So that it is possible to stop in the target

Epigraph of the minimum time function

C

convex hull of the dynamics with zero

Epigraph of the minimum time function

C

Epigraph of the minimum time function

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 PDEI. Epigraphical solutionII. 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 graphII. The set-valued viability solutionIII. 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 domainboundary

[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 PDEI. Epigraphical solutionII. 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 graphII. The set-valued viability solutionIII. A computational example

IV. Towards a selection criterion for uniqueness

Characteristic systemConsider 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 PDEI. Epigraphical solutionII. 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 graphII. The set-valued viability solutionIII. A computational example

IV. Towards a selection criterion for uniqueness

Initial conditions only

Initial and boundary conditions

Initial and boundary conditions, constraints

Initial and boundary conditions, constraints

Outline

I. The capture basin and viability kernel

II. The capture basin: an abstraction to solve a PDEI. Epigraphical solutionII. 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 graphII. The set-valued viability solutionIII. A computational example

IV. Towards a selection criterion for uniqueness

Computation with VIABILYS

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 solution Viability 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 PDEI. Epigraphical solutionII. 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 graphII. The set-valued viability solutionIII. 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 constraintsProblem: 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 systemConstant input uInitial condition x

Which x are such that after an

infinite number of iterations,

is still in the ballxj

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