A minimax optimal control problem

Holmåker, Kjell

doi:10.1007/bf00933382

Cited by 21 publications

(3 citation statements)

References 4 publications

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“…L = 0, g = 0) has been extensively studied from various point of views, including dynamic programming, numerical approximations, and necessary conditions (see e.g. [4][5][6][7][8][9][10][11][14][15][16]19,20]). In particular, in [7] Barron and Ishii established a Hamilton-Jacobi equation by regarding a L ∞ problem as the limit of standard L p optimal control problems.…”

Section: X(t) = F (T X(t) A(t)) X(τ ) = Ymentioning

confidence: 99%

(L ∞ + Bolza) control problems as dynamic differential games

Bettiol

Rampazzo

2013

Nonlinear Differ. Equ. Appl.

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Abstract. We consider a (L ∞ + Bolza) control problem, namely a problem where the payoff is the sum of a L ∞ functional and a classical Bolza functional (the latter being an integral plus an end-point functional). Owing to the L 1 , L ∞ duality, the (L ∞ +Bolza) control problem is rephrased in terms of a static differential game, where a new variable k plays the role of maximizer (we regard 1 − k as the available fuel for the maximizer). The relevant fact is that this static game is equivalent to the corresponding dynamic differential game, which allows the (upper) value function to verify a boundary value problem. This boundary value problem involves a Hamilton-Jacobi equation whose Hamiltonian is continuous. The fueled value function W(t, x, k)-whose restriction to k = 0 coincides with the value function of the reference (L ∞ + Bolza) problem-is continuous and solves the established boundary value problem. Furthermore, W is the unique viscosity solution in the class of (not necessarily continuous) bounded solutions.Mathematics Subject Classification. 49K35 minimax problems · 49N70 differential games · 49L25 viscosity solutions.

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Section: X(t) = F (T X(t) A(t)) X(τ ) = Ymentioning

confidence: 99%

(L ∞ + Bolza) control problems as dynamic differential games

Bettiol

Rampazzo

2013

Nonlinear Differ. Equ. Appl.

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“…64-65. Holmåker [13] considered a minimax problem given in control form and derived a general Pontryagin-type maximum principle; p. 395. This principle also includes transversality end conditions.…”

Section: A Basic Theoremmentioning

confidence: 99%

On certain minimax problems and Pontryagin’s maximum principle

Aronsson

2009

Calc. Var.

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This paper deals with minimax problems for nonlinear differential expressions involving a vector-valued function of a scalar variable under rather conventional structure conditions on the cost function. It is proved that an absolutely minimizing (i.e. globally and locally minimizing) function is continuously differentiable. A minimizing function is also continuously differentiable, provided a certain extra condition is satisfied. The variational method of V.G. Boltyanskii, developed within optimal control theory, is adapted and used in the proof. The case of higher order derivatives is also considered.

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“…ii{t) = x 2 (t) xi(0)=0, i j ( 0 = «(0 352(0) = 0 , terminal state constraint zi(l) = 1, 12(1) = 0. The exact analytical solution for this problem is available in [10]. The solution obtained from the present calculations using n p = 10 and arbitrary initial guess is g$ = 16.61 as compared to the exact value of ffSexact = 16.68.…”

Section: Example 2 Minimax Optimal Control Problemmentioning

confidence: 99%

A computational method for combined optimal parameter selection and optimal control problems with general constraints

Teo

Goh

1989

J. Aust. Math. Soc. Series B, Appl. Math

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In this paper, we consider a class of combined optimal parameter selection and optimal cttfitrol problems with general constraints. The first aim is to provide a unified approach to the numerical solution of this general class of optimisation problems by using the control parametrisation technique. This approach is supported by some convergence results. The second aim is to show that several different classes of optimal control problems can all be transformed into special cases of the problem considered in this paper. For illustration, four numerical examples are presented.

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A minimax optimal control problem

Cited by 21 publications

References 4 publications

(L ∞ + Bolza) control problems as dynamic differential games

(L ∞ + Bolza) control problems as dynamic differential games

On certain minimax problems and Pontryagin’s maximum principle

A computational method for combined optimal parameter selection and optimal control problems with general constraints

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