Shock Waves for Riccati Partial Differential Equations Arising in Nonlinear Optimal Control

Byrnes, Christopher I.; Jhemi, Ali

doi:10.1007/978-1-4757-2204-8_16

Cited by 7 publications

(5 citation statements)

References 3 publications

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“…Example (see [10]). Consider the one-dimensional control problem to minimize J(u) = 1 2 T 0 u 2 dt + 1 2…”

Section: The Map Is Bijective and Has An Everywhere Nonvanishing Jacomentioning

confidence: 97%

Parametrized Families of Extremals and Singularities in Solutions to the Hamilton--Jacobi--Bellman Equation

Kiefer¹,

Schättler²

1999

SIAM J. Control Optim.

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We analyze the effect which a fold and simple cusp singularity in the flow of a parametrized family of extremal trajectories of an optimal control problem has on the corresponding parametrized cost or value function. A fold singularity in the flow of extremals generates an edge of regression of the value implying the well-known results that trajectories stay strongly locally optimal until the fold-locus is reached, but lose optimality beyond. Thus fold points correspond to conjugate points. A simple cusp point in the parametrized flow of extremals generates a swallowtail in the parametrized value. More specifically, there exists a region in the state space which is covered 3:1 with both locally minimizing and maximizing branches. The changes from the locally minimizing to the maximizing branch occur at the fold-loci and there trajectories lose strong local optimality. However, the branches intersect and generate a cut-locus which limits the optimality of close-by trajectories and eliminates these trajectories from optimality near the cusp point prior to the conjugate point. In the language of partial differential equations, a simple cusp point generates a shock in the solutions to the Hamilton-Jacobi-Bellman equation while fold points will not be part of the synthesis of optimal controls near the simple cusp point. Introduction.We study singularities in solutions to the Hamilton-JacobiBellman equation for the value-function of an optimal control problem for ordinary differential equations. It is well known (see, for instance, [3]) that the necessary conditions for optimality given in the Pontryagin maximum principle [28] also give the characteristic equations for the Hamilton-Jacobi-Bellman equation. Thus, if the flow of extremals (trajectories which satisfy the necessary conditions of the Pontryagin maximum principle) covers an open region R of the state-space diffeomorphically, then a smooth solution to the Hamilton-Jacobi-Bellman equation can be constructed on R by the method of characteristics. In general, however, except for special problems like the linear-quadratic regulator, the value function is typically not smooth. The difficulties in finding solutions to optimal control problems or, equivalently, in finding solutions to the Hamilton-Jacobi-Bellman equation, precisely lie in analyzing the singularities.There is a large body of literature on singularities of solutions to the HamiltonJacobi-Bellman equation. Most of these papers deal with general topological properties of the singular set or try to establish smoothness of solutions. In [17] Fleming proves that the singular set is closed and of Hausdorff dimension at most equal to the dimension of the state space. Dafermos [16] analyzes singular sets for more general hyperbolic conservation laws, but only in dimension one. Cannarsa and Soner [11] analyze the local structure of the singular set for viscosity solutions which satisfy cer-

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“…Example (see [10]). Consider the one-dimensional control problem to minimize J(u) = 1 2 T 0 u 2 dt + 1 2…”

Section: The Map Is Bijective and Has An Everywhere Nonvanishing Jacomentioning

confidence: 97%

Parametrized Families of Extremals and Singularities in Solutions to the Hamilton--Jacobi--Bellman Equation

Kiefer¹,

Schättler²

1999

SIAM J. Control Optim.

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“…then, as for an L 1 -type objective (11), minimizing H is simple. For sake of argument (and since this is the most common case considered) suppose the functions L i are positive.…”

Section: Problems With Amentioning

confidence: 99%

“…Contrary to the belief of many authors which use this approach, formula (17) does not define an optimal control: the stationary point u i (t) depends on the multiplier λ(t) which, together with the state x * (t) is only given as the solution to a 2-point boundary value problem. There simply may exist multiple solutions and it is easy to give mathematical examples when this happens [10,11]. Geometrically, the formula (17) determines the control u * only as a function in the cotangent bundle, u * = u * (x, λ), while the optimal control u * lives in the state-space only, i.e., u * = u * (x).…”

Section: Problems With Amentioning

confidence: 99%

Pitfalls in applying optimal control to dynamical systems: An overview and editorial perspective

Ledzewicz

Schättler

2022

DCDS-B

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<p style='text-indent:20px;'>In recent years, an increasing number of papers have been published (and many more submitted for publication) in which optimal control theory is superficially applied to specific problems, especially from the biological and health sciences, but also many other fields. A lack of understanding of what it actually means to solve an optimal control problem—complex infinite-dimensional optimization problems—often leads to heavily overblown claims about optimality of solutions. In this editorial, a critical assessment of these efforts is given.</p>

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“…R n : We are interested in long-time existence of solutions on arbitrary intervals ½t 0 ; T; with the obstructions to existence being finite escape time or the existence of a shock wave at some time t: We shall first define a variational problem for which, when viewed as an optimal control problem, the Riccati partial differential equation [15] for the gradient of the value function turns out to be the vector inviscid Burgers' equation in backwards time.…”

Section: The Absence Of Shock Waves In Burgers' Equationmentioning

confidence: 99%

“…We begin by defining a variational problem for which, when viewed as an optimal control problem, the Riccati partial differential equation [15] for the gradient of the value function turns out to be the Burgers' equation in backwards time. We then construct, for each t 2 ½t 0 ; T; a map f t for which well-posedness is equivalent to the absence of shock waves at time t: Several criteria for properness of this map are given in terms of qualitative behaviour of the initial data.…”

Section: Introductionmentioning

confidence: 99%

Interior point solutions of variational problems and global inverse function theorems

Byrnes

Lindquist

2006

Intl J Robust & Nonlinear

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SUMMARYVariational problems and the solvability of certain nonlinear equations have a long and rich history beginning with calculus and extending through the calculus of variations. In this paper, we are interested in 'well-connected' pairs of such problems which are not necessarily related by critical point considerations. We also study constrained problems of the kind which arise in mathematical programming. We are also interested in interior minimizing points for the variational problem and in the well-posedness (in the sense of Hadamard) of solvability of the related systems of equations. We first prove a general result which implies the existence of interior points and which also leads to the development of certain generalization of the Hadamard-type global inverse function theorem, along the theme that uniqueness quite often implies existence. This result is illustrated by proving the non-existence of shock waves for certain initial data for the vector Burgers' equation. The global inverse function theorem is also illustrated by a derivation of the existence of positive definite solutions of matrix Riccati equations without first analysing the nonlinear matrix Riccati differential equation. The main results on the existence of solutions to geometrically constrained well-connected pairs are then presented and illustrated by a geometric analysis of the existence of interior points for linear programming problems.

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Shock Waves for Riccati Partial Differential Equations Arising in Nonlinear Optimal Control

Cited by 7 publications

References 3 publications

Parametrized Families of Extremals and Singularities in Solutions to the Hamilton--Jacobi--Bellman Equation

Parametrized Families of Extremals and Singularities in Solutions to the Hamilton--Jacobi--Bellman Equation

Pitfalls in applying optimal control to dynamical systems: An overview and editorial perspective

Interior point solutions of variational problems and global inverse function theorems

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