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(1990)
Basel: Birkhäuser, Basel · Zbl 0713
L. Weiss (1970)
Introduction to the mathematical theory of control processes, Vol. I - Linear equations and quadratic criteriaIEEE Transactions on Automatic Control, 15
(1977)
Vector Measures (1977), Providence: American Mathematical Society, Providence
F. Santambrogio (2015)
Optimal Transport for Applied Mathematicians
M. Fornasier, Francesco Solombrino (2013)
Mean-Field Optimal ControlESAIM: Control, Optimisation and Calculus of Variations, 20
Z. Denkowski, S. Migórski, N. Papageorgiou (2021)
Set-Valued Analysis
M. Chambers (1965)
The Mathematical Theory of Optimal ProcessesJournal of the Operational Research Society, 16
Giulia Cavagnari, S. Lisini, Carlo Orrieri, Giuseppe Savaré (2020)
Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergenceJournal of Differential Equations
M. Fornasier, S. Lisini, Carlo Orrieri, Giuseppe Savaré (2018)
Mean-field optimal control as Gamma-limit of finite agent controlsEuropean Journal of Applied Mathematics, 30
C. Jimenez, A. Marigonda, M. Quincampoix (2020)
Optimal control of multiagent systems in the Wasserstein spaceCalculus of Variations and Partial Differential Equations, 59
C. Villani (2008)
Optimal Transport: Old and New
R. Carmona, F. Delarue (2018)
Probabilistic Theory of Mean Field Games with Applications II: Mean Field Games with Common Noise and Master Equations
Giulia Cavagnari, A. Marigonda, B. Piccoli (2017)
Superposition Principle for Differential Inclusions
R. Carmona, F. Delarue (2013)
Forward-Backward Stochastic Differential Equations and Controlled McKean Vlasov DynamicsAnnals of Probability, 43
H. Berg (2019)
Differential inclusionsHormones as Tokens of Selection
M. Bongini, M. Fornasier, Francesco Rossi, Francesco Solombrino (2015)
Mean-Field Pontryagin Maximum PrincipleJournal of Optimization Theory and Applications, 175
N. Bellomo, M. Herrero, A. Tosin (2012)
On the dynamics of social conflicts: looking for the Black SwanArXiv, abs/1202.4554
Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors
H. Frankowska, N. Osmolovskii (2019)
Second-Order Necessary Conditions for a Strong Local Minimum in a Control Problem with General Control ConstraintsApplied Mathematics & Optimization
H. Frankowska (1987)
The maximum principle for an optimal solution to a differential inclusion with end points constraintsSiam Journal on Control and Optimization, 25
(2006)
Handbook of Differential Equations: Evolutionary Equations, vol
F Otto (2001)
The geometry of dissipative equations: the porous medium equationCommun. Partial Differ. Equ., 26
H. Frankowska, N. Osmolovskii (2020)
Distance estimates to feasible controls for systems with final point constraints and second order necessary optimality conditionsSyst. Control. Lett., 144
A. Muntean, J. Rademacher, A. Zagaris (2016)
Macroscopic and large scale phenomena : coarse graining, mean field limits and ergodicity, 3
B. Piccoli, Francesco Rossi, E. Trélat (2014)
Control to Flocking of the Kinetic Cucker-Smale ModelSIAM J. Math. Anal., 47
H. Frankowska (1990)
A priori estimates for operational differential inclusionsJournal of Differential Equations, 84
B. Piccoli, Francesco Rossi (2011)
Transport Equation with Nonlocal Velocity in Wasserstein Spaces: Convergence of Numerical SchemesActa Applicandae Mathematicae, 124
L. Ambrosio, D. Pallara, N. Fusco (2000)
Functions of Bounded Variation and Free Discontinuity Problems
E. Cristiani, B. Piccoli, A. Tosin (2014)
Multiscale Modeling of Pedestrian Dynamics
R. Bellman, L. Pontryagin, V. Boltyanskii, R. Gamkrelidze, E. Mischenko (1962)
Mathematical Theory of Optimal Processes
F. Bullo (2009)
Distributed Control of Robotic Networks
L. Ambrosio, N. Gigli, Giuseppe Savaré (2005)
Gradient Flows: In Metric Spaces and in the Space of Probability Measures
R. McCann (1997)
A Convexity Principle for Interacting GasesAdvances in Mathematics, 128
F. Bullo, J. Cortés, S. Martínez (2009)
Distributed Control of Robotic Networks: A Mathematical Approach to Motion Coordination Algorithms
L. Ambrosio, N. Gigli (2008)
Construction of the Parallel Transport in the Wasserstein SpaceMethods and applications of analysis, 15
H. Frankowska, N. Osmolovskii (2018)
Strong Local Minimizers in Optimal Control Problems with State Constraints: Second-Order Necessary ConditionsSIAM J. Control. Optim., 56
Benoît Bonnet, H. Frankowska (2020)
Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz frameworkJournal of Differential Equations
G. Albi, M. Bongini, E. Cristiani, D. Kalise (2015)
Invisible Control of Self-Organizing Agents Leaving Unknown EnvironmentsSIAM J. Appl. Math., 76
M. Fornasier, B. Piccoli, Francesco Rossi (2014)
Mean-field sparse optimal controlPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372
N. Pogodaev, M. Staritsyn (2019)
Impulsive control of nonlocal transport equationsJournal of Differential Equations
F. Santambrogio (2015)
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling
R. Carmona, F. Delarue (2018)
Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games
N. Pogodaev (2015)
Optimal control of continuity equationsNonlinear Differential Equations and Applications NoDEA, 23
Benoît Bonnet, Francesco Rossi (2019)
Intrinsic Lipschitz Regularity of Mean-Field Optimal ControlsSIAM J. Control. Optim., 59
J. Carrillo, Young-Pil Choi, M. Hauray (2013)
The derivation of swarming models: Mean-field limit and Wasserstein distancesarXiv: Analysis of PDEs, 553
M. Caponigro, B. Piccoli, Francesco Rossi, Emmanuel Tr'elat (2017)
Mean-field sparse Jurdjevic-Quinn controlMathematical Models and Methods in Applied Sciences, 27
G. Albi, L. Pareschi, M. Zanella (2014)
Boltzmann-type control of opinion consensus through leadersPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372
R. Bellman (1967)
Introduction to the mathematical theory of control processes
J. Aubin, A. Cellina (1984)
Differential Inclusions - Set-Valued Maps and Viability Theory, 264
W. Gangbo, A. Tudorascu (2019)
On differentiability in the Wasserstein space and well-posedness for Hamilton–Jacobi equationsJournal de Mathématiques Pures et Appliquées
F. Cucker, S. Smale (2007)
On the mathematics of emergenceJapanese Journal of Mathematics, 2
(1977)
Vector Measures, vol
H. Frankowska, E. Marchini, M. Mazzola (2018)
Necessary optimality conditions for infinite dimensional state constrained control problemsJournal of Differential Equations
H. Frankowska, Qi Lu (2019)
First and second order necessary optimality conditions for controlled stochastic evolution equations with control and state constraintsJournal of Differential Equations
J. Lasry, P. Lions (2007)
Mean field gamesJapanese Journal of Mathematics, 2
Benoît Bonnet (2018)
A Pontryagin Maximum Principle in Wasserstein spaces for constrained optimal control problemsESAIM: Control, Optimisation and Calculus of Variations
H. Frankowska, Haisen Zhang, Xu Zhang (2018)
Stochastic Optimal Control Problems with Control and Initial-Final States ConstraintsSIAM J. Control. Optim., 56
Benoît Bonnet, Francesco Rossi (2017)
The Pontryagin Maximum Principle in the Wasserstein SpaceCalculus of Variations and Partial Differential Equations, 58
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F. Otto (2001)
THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATIONCommunications in Partial Differential Equations, 26
In this article, we derive first-order necessary optimality conditions for a constrained optimal control problem formulated in the Wasserstein space of probability measures. To this end, we introduce a new notion of localised metric subdifferential for compactly supported probability measures, and investigate the intrinsic linearised Cauchy problems associated to non-local continuity equations. In particular, we show that when the velocity perturbations belong to the tangent cone to the convexification of the set of admissible velocities, the solutions of these linearised problems are tangent to the solution set of the corresponding continuity inclusion. We then make use of these novel concepts to provide a synthetic and geometric proof of the celebrated Pontryagin Maximum Principle for an optimal control problem with inequality final-point constraints. In addition, we propose sufficient conditions ensuring the normality of the maximum principle.
Applied Mathematics and Optimization – Springer Journals
Published: Dec 1, 2021
Keywords: Mean-field optimal control; Wasserstein spaces; Pontryagin maximum principle; Differential inclusions; Inner-approximations of optimal trajectories; 30L99; 34K09; 49J53; 49K21; 49Q22; 58E25
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