Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

Bokanowski, Olivier; Désilles, Anya; Zidani, Hasnaa

doi:10.1051/cocv/2021084

Cited by 7 publications

(4 citation statements)

References 25 publications

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“…Many dynamic optimization methods can be used to solve this control problem, including Pontryagin's maximum principle and Bellman's optimality principle. The differences between these methods were described by Bokanowski et al in [31].…”

Section: Optimization Of Safe Trajectorymentioning

confidence: 99%

Creating Autonomous Multi-Object Safe Control via Different Forms of Neural Constraints of Dynamic Programming

Lisowski

2024

Electronics

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The aim of this work, which is an extension of previous research, is a comparative analysis of the results of the dynamic optimization of safe multi-object control, with different representations of the constraints of process state variables. These constraints are generated with an artificial neural network and take movable shapes in the form of a parabola, ellipse, hexagon, and circle. The developed algorithm allows one to determine a safe and optimal trajectory of an object when passing other multi-objects. The obtained results of the simulation tests of the algorithm allow for the selection of the best representation of the motion of passing objects in the form of neural constraints. Moreover, the obtained characteristics of the sensitivity of the object’s trajectory to the inaccuracy of the input data make it possible to select the best representation of the motion of other objects in the form of an excessive approximation area as neural constraints of the control process.

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Section: Optimization Of Safe Trajectorymentioning

confidence: 99%

Creating Autonomous Multi-Object Safe Control via Different Forms of Neural Constraints of Dynamic Programming

Lisowski

2024

Electronics

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“…Our Hopf-Lax formulae in Section V will assume convex Hamiltonians in the gradient space. H (12) and HTI 2 (20) are convex in (p, q), but HTI 1 (14) is not. Thus, we do not have the Hopf-Lax formula for the time-invariant SCCIP.…”

Section: B Hamilton-jacobi Equation For Scrapmentioning

confidence: 99%

“…Now, it is sufficient to show V 2 ≡ W TI 2 . To prove this, we will use Theorem 3 where we substitute (20) to F 1 and (56) to F 2 .…”

Section: B Hopf-lax Formula For Scrapmentioning

confidence: 99%

Efficient Computation of State-Constrained Reachability Problems Using Hopf–Lax Formulae

Lee

Tomlin

2023

IEEE Trans. Automat. Contr.

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This paper considers two state-constrained reachability problems: computing (1) control-invariant and(2) reach-avoid sets, both under state constraints. Prior research has developed Hamilton-Jacobi (HJ) partial differential equations (PDEs) that characterize the optimal cost functions of these two problems. Unfortunately, solving the HJ PDEs by grid-based methods, such as levelset methods, suffers from exponentially growing computational complexity in the system state dimension. In order to alleviate this computational issue, this paper proposes a Hopf-Lax formula for each reachability problem's HJ PDE. The advantage of the Hopf-Lax formulae is that they have more favorable convexity conditions than the corresponding problems. Thus, direct methods may be used to solve Hopf-Lax formulae and thus efficiently compute the optimal solution of the reachability problems under specified conditions. This paper provides an example for each reachability problem and demonstrates the performance of the proposed Hopf-Lax method.

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“…In this work, we consider a general control problem with pathwise state inequality constraints. We follow some ideas introduced in [1,10] and reformulate the original control problem as a control problem with a maximum running cost whose value function is locally Lipschitz continuous everywhere. Then, by using the results that link the original control problem with the auxiliary problem, we are able to derive a set of sensitivity relations without requiring any type of compatibility assumptions such as the mentioned above.…”

mentioning

confidence: 99%

Relationship Between the Maximum Principle and Dynamic Programming for Minimax Problems

Hermosilla

Zidani²

2023

Appl Math Optim

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This paper is concerned with the relationship between the maximum principle and dynamic programming for a large class of optimal control problems with maximum running cost. Inspired by a technique introduced by Vinter in the 1980s, we are able to obtain jointly a global and a partial sensitivity relation that link the coextremal with the value function of the problem at hand. One of the main contributions of this work is that these relations are derived by using a single perturbed problem, and therefore, both sensitivity relations hold, at the same time, for the same coextremal.As a by-product, and thanks to the level-set approach, we obtain a new set of sensitivity relations for Mayer problems with state constraints. One important feature of this last result is that it holds under mild assumptions, without the need of imposing strong compatibility assumptions between the dynamics and the state constraints set. Minimax optimal control problems and Maximum principle and Dynamic programming and Sensitivity analysis and State constraints

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Relationship between maximum principle and dynamic programming in presence of intermediate and final state constraints

Cited by 7 publications

References 25 publications

Creating Autonomous Multi-Object Safe Control via Different Forms of Neural Constraints of Dynamic Programming

Creating Autonomous Multi-Object Safe Control via Different Forms of Neural Constraints of Dynamic Programming

Efficient Computation of State-Constrained Reachability Problems Using Hopf–Lax Formulae

Relationship Between the Maximum Principle and Dynamic Programming for Minimax Problems

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