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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
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