Theory and applications of optimal control problems with multiple time-delays
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Cites background or methods from "Theory and applications of optimal ..."
...], λ˙L 1(t) = −HL1[t], λ˙L 2(t) = −HL2[t], and λ˙I(t) = −HI[t]+χ[0,T−d I]Hy3[t+dI], where subscripts denote partial derivatives and χ[0,T−d I] is the characteristic function in the interval [0,T −dI] [10]. Note that only the equation for λ˙ I(t) contains the advanced time t+ dI. Since the terminal state x(T) is free, the transversality conditions are λS(T) = λL1(T) = λI(T) = λL2(T) = 0. (20) To charac...
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...owerful optimization solvers. We use the Interior-Point optimization solver IPOPT developed by Wachter and Biegler [30]. Details of discretization methods for delayed control problems may be found in [10]. The subsequent computations for the terminal time tf = 5 have been performed with N = 2500 to N = 5000 grid points using the trapezoidal rule as integration method. Choosing the error tolerance tol ...
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...15), initial conditions (16) and control constraints (17). Necessary optimality conditions for optimal control problems with multiple time delays in control and state variables may be found, e.g., in [10]. Here, we discuss the Maximum Principle in order to display the controls and the switching functions in a convenient way. To define the Hamiltonian forthe delayedcontrolproblem, we introduce the delay...
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51 citations
39 citations
Cites background or methods from "Theory and applications of optimal ..."
...A necessary condition for optimal control problems can be found in [13] and it’s applications [11, 12]....
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...Then, we consider a range of issues related to the delay optimal control with the method of Pontryagin’s Maximum Principle similar to the way as what the documents [11, 12] did....
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39 citations
Cites background from "Theory and applications of optimal ..."
...In particular, Many theoretical results are now available in the literature, which include necessary optimality conditions for time-delay optimal control problems [5,7]....
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39 citations
References
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...[33, 34]....
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6,056 citations
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...[14, 20, 22, 25] assures the existence of a adjoint (costate) function Λ̂ ∈W 1,∞([0, h],R ·n), a multiplier λ0 ≥ 0, a multiplier function M̂ ∈ L∞([0, h],R ·p) and a vector ν̂ ∈ R(N−1)·n+q, ν̂ = (ν̂∗ 0 , ....
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3,176 citations
"Theory and applications of optimal ..." refers methods in this paper
...To solve the optimization problem (NLP) in (40) – (43) numerically, we employ the programming language AMPL developed by Fourer, Gay and Kernighan [9] in conjunction with the optimization solvers LOQO by Vanderbei [31, 32] or IPOPT by Wächter et al....
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1,127 citations
"Theory and applications of optimal ..." refers background in this paper
...[1, 3, 6, 7, 11, 13, 15, 16, 17, 21, 23, 26, 27, 28, 35]....
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...In particular, Warga [35] presents a general approach for deriving necessary optimality conditions which is based on optimization theory in function spaces....
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...We must admit that we could not translate the necessary conditions of Warga [35] in the presence of a practical example like the one in Section 6....
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1,021 citations
Additional excerpts
...[14, 20, 22, 25] assures the existence of a adjoint (costate) function Λ̂ ∈W 1,∞([0, h],R ·n), a multiplier λ0 ≥ 0, a multiplier function M̂ ∈ L∞([0, h],R ·p) and a vector ν̂ ∈ R(N−1)·n+q, ν̂ = (ν̂∗ 0 , ....
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