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Bonnans, F.; Martinon, P.; Trélat, E.
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25 Citations
We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control.
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Sethi, S. P.; Prasad, A.; He, X.
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36 Citations
A model of newproduct adoption is proposed that incorporates price and advertising effects. An optimal control problem that uses the model as its dynamics is solved explicitly to obtain the optimal price and advertising effort over time. The model has a great potential to be used in obtaining solutions and insights in a variety of differential game settings.
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Dmitruk, A. V.; Vdovina, A. K.
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A onedimensional optimal control problem with a statedependent cost and a unimodular integrand is considered. It is shown that, under some standard assumptions, this problem can be solved without using the Pontryagin maximum principle, by simple methods of the classical analysis, basing on the Tchyaplygin comparison theorem. However, in some modifications of the problem, the usage of Pontryagin’s maximum principle is preferable. The optimal synthesis for the problem and for its modifications is obtained.
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Bergounioux, M.
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10 Citations
We investigate optimal control problems governed by variational inequalities involving constraints on the control, and more precisely the example of the obstacle problem. In this paper, we discuss some augmented Lagrangian algorithms to compute the solution.
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Carlson, D. A.
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5 Citations
In this paper, we extend the existence theory of Brock and Haurie concerning the existence of sporadically catchingup optimal solutions for autonomous, infinitehorizon optimal control problems. This notion of optimality is one of a hierarchy of types of optimality that have appeared in the literature to deal with optimal control problems whose cost functionals, described by an improper integral, either diverge or are unbounded below. Our results rely on the now classical convexity and seminormality hypotheses due to Cesari and are weaker than those assumed in the work of Brock and Haurie. An example is presented where our results are applicable, but those of the abovementioned authors do not.
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Fuhrman, Marco; Hu, Ying; Tessitore, Gianmario
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12 Citations
We prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a stochastic partial differential equation driven by a finite dimensional Wiener process. The equation is formulated in a semiabstract form that allows direct applications to a large class of controlled stochastic parabolic equations. We allow for a diffusion coefficient dependent on the control parameter, and the space of control actions is general, so that in particular we need to introduce two adjoint processes. The second adjoint process takes values in a suitable space of operators on L^{4}.
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Preininger, J.; Vuong, P. T.
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2 Citations
We revisit the gradient projection method in the framework of nonlinear optimal control problems with bang–bang solutions. We obtain the strong convergence of the iterative sequence of controls and the corresponding trajectories. Moreover, we establish a convergence rate, depending on a constant appearing in the corresponding switching function and prove that this convergence rate estimate is sharp. Some numerical illustrations are reported confirming the theoretical results.
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Sager, Sebastian; Bock, Hans Georg; Reinelt, Gerhard
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45 Citations
Many practical optimal control problems include discrete decisions. These may be either timeindependent parameters or timedependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixedinteger linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixedinteger programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixedinteger optimal control problems, with a focus on discretevalued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.
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Bergounioux, M.
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16 Citations
We consider stateconstrained optimal control problems governed by elliptic equations. Doing Slaterlike assumptions, we know that Lagrange multipliers exist for such problems, and we propose a decoupled augmented Lagrangian method. We present the algorithm with a simple example of a distributed control problem.
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Bonaccorsi, Stefano; Mastrogiacomo, Elisa
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5 Citations
We apply the semigroup setting of Desch and Miller to a class of stochastic integral equations of Volterra type with completely monotone kernels with a multiplicative noise term; the corresponding equation is an infinite dimensional stochastic equation with unbounded diffusion operator that we solve with the semigroup approach of Da Prato and Zabczyk. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise.
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