A dichotomy in linear control theory

Wilde, R.W.; Kokotović, P.V.

doi:10.1109/tac.1972.1099976

Cited by 50 publications

(37 citation statements)

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“…Based on Hamilton-Jacobi approach, the turnpike property is proved in [38] for linear quadratic problems under the Kalman condition (see [33] of Chapter 5 for a definition of the Kalman condition), and for the nonlinear control-affine system in [1]. Finally, in [23] they analyse the convergence of the optimal solution to the steady state in large time problems.…”

Section: The Turnpike Theorymentioning

confidence: 99%

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

Ibañez

2016

Journal of Biological Dynamics

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The Lotka-Volterra model is a differential system of two coupled equations representing the interaction of two species: a prey one and a predator one. We formulate an optimal control problem adding the effect of hunting both species as the control variable. We analyse the optimal hunting problem paying special attention to the nature of the optimal state and control trajectories in long time intervals. To do that, we apply recent theoretical results on the frame to show that, when the time horizon is large enough, optimal strategies are nearly steady-state. Such path is known as turnpike property. Some experiments are performed to observe such turnpike phenomenon in the hunting problem. Based on the turnpike property, we implement a variant of the single shooting method to solve the previous optimisation problem, taking the middle of the time interval as starting point. ARTICLE HISTORY

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Section: The Turnpike Theorymentioning

confidence: 99%

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

Ibañez

2016

Journal of Biological Dynamics

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“…Specifically, the proposed approach is based on turnpike theory [21,22] in optimal control, which analyzes optimal control problems with respect to the structure of the optimal solution, i.e., optimal trajectories of the control and state variables. Turnpike theory has been used in the context of indirect methods in [23][24][25], where the structure of the optimal trajectories is exploited to approximate the resulting boundary-value problem by two initial-value problems corresponding to the initial and terminal segments of the time horizon. However, the focus in this work is on using turnpike theory for efficient control discretization in direct methods.…”

Section: Introductionmentioning

confidence: 99%

Efficient Control Discretization Based on Turnpike Theory for Dynamic Optimization

Sahlodin

Barton

2017

Processes

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Dynamic optimization offers a great potential for maximizing performance of continuous processes from startup to shutdown by obtaining optimal trajectories for the control variables. However, numerical procedures for dynamic optimization can become prohibitively costly upon a sufficiently fine discretization of control trajectories, especially for large-scale dynamic process models. On the other hand, a coarse discretization of control trajectories is often incapable of representing the optimal solution, thereby leading to reduced performance. In this paper, a new control discretization approach for dynamic optimization of continuous processes is proposed. It builds upon turnpike theory in optimal control and exploits the solution structure for constructing the optimal trajectories and adaptively deciding the locations of the control discretization points. As a result, the proposed approach can potentially yield the same, or even improved, optimal solution with a coarser discretization than a conventional uniform discretization approach. It is shown via case studies that using the proposed approach can reduce the cost of dynamic optimization significantly, mainly due to introducing fewer optimization variables and cheaper sensitivity calculations during integration.

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“…Dichotomic transformations have been used to solve boundary-value problems for ordinary differential equations [4] and to solve fixed end-point optimal control problems [2], [20]. Chow used dichotomic transformations to solve nonlinear HBVPs with linear boundary-layer dynamics [8].…”

Section: Introductionmentioning

confidence: 99%

Approximate solution of hyper-sensitive optimal control problems using finite-time Lyapunov analysis

Aykutlug

Mease

2009

2009 American Control Conference

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Abstract-Solving optimal control problems by an indirect method is often abandoned in favor of a direct method due to hyper-sensitivity with respect to unknown boundary conditions for the Hamiltonian boundary-value problem that represents the first-order necessary conditions. Yet the hyper-sensitivity may imply a manifold structure for the flow in the Hamiltonian phase space, structure that provides insight regarding the optimal solutions and suggests a solution approximation strategy that avoids the hyper-sensitivity. This paper concerns the development of a solution approximation method based on finite-time Lyapunov exponents and vectors. The focus is on determining the unknown boundary conditions such that the solution end points lie on certain invariant manifolds. Using kinematic eigenvalues, a systematic approach to determine the appropriate finite-time for the Lyapunov analysis is presented. A simple example is used to illustrate the approximation method and its implementation.

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A dichotomy in linear control theory

Cited by 50 publications

References 1 publication

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

Optimal control of the Lotka–Volterra system: turnpike property and numerical simulations

Efficient Control Discretization Based on Turnpike Theory for Dynamic Optimization

Approximate solution of hyper-sensitive optimal control problems using finite-time Lyapunov analysis

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