A review of gradient algorithms for numerical computation of optimal trajectories

Golfetto, Wander Almodovar; Fernandes, Sandro da Silva

doi:10.5028/jatm.2012.04020512

Cited by 14 publications

(13 citation statements)

References 22 publications

(35 reference statements)

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“…The second order method presented here can be applied for numerical solution of various physical and engineering problems. For example, it allows to find optimal trajectories for nonlinear dynamical systems, particularly, for aircraft [Miele, 1958;Gorbunov and Lutoshkin, 2004], spacecraft [Miele, 1958;Breakwell, 1962;Golfetto and Silva, 2012], and robots. Among other problems where this method can be used, it should be mentioned the problem of optimization of charged particle beam accelerator channel [Ovsyannikov, 1980;Bublik, Garashchenko, and Kirichenko, 1985;Drivotin et al, 1998;Ovsyannikov and Drivotin, 2003;Drivotin and Vlasova, 2014;Altsybeev et al;Drivotin, 2018].…”

Section: Resultsmentioning

confidence: 99%

“…A method accounting also the second variation of the cost functional is a second order method. The commonly used expression for the second variation of the cost functional [Longmuir and Bohn, 1969;Gabasov and Kirillova, 1973;Miele, 1975;Golfetto and Silva, 2012] contains the first variation of the trajectory δx and the variation of the control function δu:…”

Section: Introductionmentioning

confidence: 99%

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Second variation of trajectory of a controlled dynamical system

Drivotin

2018

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An integral representation for the second variation of trajectory of a dynamical system under control is obtained. This representation contains some tensor of the third rank introduced here. A differential equation for this tensor is presented. A second order method for solution of the optimal control problem based on the second variation of a trajectory is proposed.Key words optimal control problem, second variation of trajectory, numerical methods of optimal control, second order method.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Second variation of trajectory of a controlled dynamical system

Drivotin

2018

CAP

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“…However, the second-order gradient algorithm is much more sensitive to the initial estimate, and thus has a difficult time even beginning to iterate towards an optimal solution (Bryson & Ho 1975) as the region of convergence is much smaller. One alternative proposal is to combine both the first-order and the second-order gradient algorithms in order to maximize the first-order’s ability to converge quickly at the beginning, with the second-order’s ability to converge more accurately at the end (Golfetto & Fernandes 2012). …”

Section: Discussionmentioning

confidence: 99%

“…Gradient-based algorithms solve the optimization problem directly without first deriving a functional with defined boundary conditions. Recent success using this approach has been achieved for solving complex control problems in mechanics (Aghababa, Amrollahi, & Borjkhani 2012; Golfetto & Fernandes 2012; Raivo 2000), epidemiology (Gupta & Rink 1973), game theory (Doležal 1978), kinetics (Lee 1964) and immunology (Joshi 2002; Kepler & Perelson 1993; Kirschner, Lenhart, & Serbin 1997). …”

Section: Introductionmentioning

confidence: 99%

Switching neuronal state: optimal stimuli revealed using a stochastically-seeded gradient algorithm

Chang

Paydarfar

2014

J Comput Neurosci

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Inducing a switch in neuronal state using energy optimal stimuli is relevant to a variety of problems in neuroscience. Analytical techniques from optimal control theory can identify such stimuli; however, solutions to the optimization problem using indirect variational approaches can be elusive in models that describe neuronal behavior. Here we develop and apply a direct gradient-based optimization algorithm to find stimulus waveforms that elicit a change in neuronal state while minimizing energy usage. We analyze standard models of neuronal behavior, the Hodgkin-Huxley and FitzHugh-Nagumo models, to show that the gradient-based algorithm: 1) enables automated exploration of a wide solution space, using stochastically generated initial waveforms that converge to multiple locally optimal solutions; and 2) finds optimal stimulus waveforms that achieve a physiological outcome condition, without a priori knowledge of the optimal terminal condition of all state variables. Analysis of biological systems using stochastically-seeded gradient methods can reveal salient dynamical mechanisms underlying the optimal control of system behavior. The gradient algorithm may also have practical applications in future work, for example, finding energy optimal waveforms for therapeutic neural stimulation that minimizes power usage and diminishes off-target effects and damage to neighboring tissue.

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“…In this article, we use HMP‐based method to solve HSSOCP. An extended version of the gradient method for solving TPBVP 54‐59 is used to solve a MPBVP which results from HMP.…”

Section: Introductionmentioning

confidence: 99%

A gradient algorithm for solution of the optimal control problem for hybrid switching systems

Salehi

Tavassoli

2020

Optim Control Appl Methods

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Summary In this article, an algorithm is presented for solving the optimal control problem for the general form of a hybrid switching system. The cost function comprises terminal, running and switching costs. The controlled system is an autonomous hybrid switching system with jumps either at some switching times or some time varying switching manifolds. The proposed algorithm is an extension of the first‐order gradient method for the conventional optimal control problem. The algorithm requires a low computational effort. The system's dynamical equations together with a set of algebraic equations are solved at each iteration in order to find the descent direction. The convergence of algorithm is proved and examples are provided to demonstrate the efficiency of the algorithm for different types of hybrid switching system optimal control problems.

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A review of gradient algorithms for numerical computation of optimal trajectories

Cited by 14 publications

References 22 publications

Second variation of trajectory of a controlled dynamical system

Second variation of trajectory of a controlled dynamical system

Switching neuronal state: optimal stimuli revealed using a stochastically-seeded gradient algorithm

A gradient algorithm for solution of the optimal control problem for hybrid switching systems

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