scholarly journals A sequential quadratic Hamiltonian scheme to compute optimal relaxed controls

2021 ◽  
Author(s):  
Alfio Borzi ◽  
Mario Annunziato
Keyword(s):  
Optimal Control Problems ◽  
Successive Approximations ◽  
Young Measures ◽  
Optimal Controls ◽  
Relaxed Controls ◽  
Concentration Effects ◽  
Convergence Results ◽  
The Family ◽  
Quadratic Hamiltonian ◽  
The Pontryagin Maximum Principle

A new sequential quadratic Hamiltonian method for computing optimal relaxed controls for a class of optimal control problems governed by ordinary differential equations is presented. This iterative approach is based on the characterisation of optimal controls by means of the Pontryagin maximum principle in the framework of Young measures, and it belongs to the family of successive approximations schemes. The ability of the proposed optimisation framework to solve problems with regular and relaxed controls, including cases with oscillations and concentration effects, is demonstrated by results of numerical experiments. In all cases, the sequential quadratic Hamiltonian scheme appears robust and efficient, in agreement with convergence results of the theoretical investigation presented in this paper.

scholarly journals Existence of optimal controls for semilinear elliptic equations without Cesari-type conditions

2003 ◽  
Vol 45 (1) ◽  
pp. 115-131 ◽  
Author(s):  
Hongwei Lou
Keyword(s):  
Existence Theorem ◽  
Elliptic Equations ◽  
Original Problem ◽  
Optimal Control Problems ◽  
Elliptic Partial Differential Equations ◽  
Control Problems ◽  
Optimal Controls ◽  
State Control ◽  
Semilinear Elliptic ◽  
The Pontryagin Maximum Principle

AbstractOptimal control problems governed by semilinear elliptic partial differential equations are considered. No Cesari-type conditions are assumed. By proving an existence theorem and the Pontryagin maximum principle of optimal “state-control” pairs for the corresponding relaxed problems, we establish an existence theorem of optimal pairs for the original problem.

scholarly journals On the SQH Method for Solving Differential Nash Games

2021 ◽  
Author(s):  
Francesca Calà Campana ◽  
Alfio Borzì
Keyword(s):  
Numerical Experiments ◽  
Optimal Control Problems ◽  
Open Loop ◽  
Successive Approximations ◽  
Control Problems ◽  
Well Posedness ◽  
Nash Games ◽  
Computational Performance ◽  
Theoretical Results ◽  
The Pontryagin Maximum Principle

AbstractA sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed scheme, and results of numerical experiments are reported that successfully validate its computational performance.

Some geometric properties of the set of generalized Young functionals

1999 ◽  
Vol 129 (3) ◽  
pp. 601-616
Author(s):  
Martin Kružík ◽  
Tomáš Roubíček
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Extreme Points ◽  
Existence Results ◽  
Young Measures ◽  
Control Problems ◽  
Geometric Properties ◽  
Convergence Results ◽  
Choquet Theory

This paper studies geometric properties, in particular extreme points and rays, of various generalizations of Young measures. Applications of the knowledge of extreme points are illustrated on existence results for optimal control problems and on various convergence results for Young measures by using the Choquet theory.

scholarly journals Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

scholarly journals An entropy minimization approach to second-order variational mean-field games

2019 ◽  
Vol 29 (08) ◽  
pp. 1553-1583 ◽  
Author(s):  
Jean-David Benamou ◽  
Guillaume Carlier ◽  
Simone Di Marino ◽  
Luca Nenna
Keyword(s):  
Efficient Algorithm ◽  
Mean Field ◽  
Time Discretization ◽  
Second Order ◽  
Mean Field Games ◽  
Time Step ◽  
Entropy Minimization ◽  
Convergence Results ◽  
Quadratic Hamiltonian ◽  
Minimization Approach

We propose an entropy minimization viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We carefully analyze the time discretization of such problems, establish [Formula: see text]-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.

Existence of optimal controls for systems of controlled forward-backward doubly SDEs

2020 ◽  
Vol 28 (2) ◽  
pp. 93-112
Author(s):  
Abdelhakim Ninouh ◽  
Boulakhras Gherbal ◽  
Nassima Berrouis
Keyword(s):  
Weak Convergence ◽  
Control Problem ◽  
Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Optimal Controls ◽  
Convexity Condition ◽  
Relaxed Controls ◽  
Relaxed Control ◽  
Doubly Stochastic ◽  
Existence Of Optimal Controls

AbstractWe wish to study a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). Firstly, we prove existence of optimal relaxed controls, which are measure-valued processes for nonlinear FBDSDEs, by using some tightness properties and weak convergence techniques on the space of Skorokhod {\mathbb{D}} equipped with the S-topology of Jakubowski. Moreover, when the Roxin-type convexity condition is fulfilled, we prove that the optimal relaxed control is in fact strict. Secondly, we prove the existence of a strong optimal controls for a linear forward-backward doubly SDEs. Furthermore, we establish necessary as well as sufficient optimality conditions for a control problem of this kind of systems. This is the first theorem of existence of optimal controls that covers the forward-backward doubly systems.

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