Optimal control problems with multiple characteristic time points in the objective and constraints

Loxton, Ryan; Teo, Kok Lay; Rehbock, Volker

doi:10.1016/j.automatica.2008.04.011

Cited by 109 publications

(111 citation statements)

References 11 publications

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“…Since (42) and (43) are implicit, rather than explicit, functions of σ and ζ, computing their gradients is a non-trivial task. We now show that these gradients can be computed using the sensitivity method described in [13].…”

Section: Solving Problemmentioning

confidence: 99%

Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

Ren

Lin

et al. 2016

Communications in Nonlinear Science and Numerical Simulation

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Establishing a good current spatial profile in tokamak fusion reactors is crucial to effective steadystate operation. The evolution of the current spatial profile is related to the evolution of the poloidal magnetic flux, which can be modeled in the normalized cylindrical coordinates using a parabolic partial differential equation (PDE) called the magnetic diffusion equation. In this paper, we consider the dynamic optimization problem of attaining the best possible current spatial profile during the rampup phase of the tokamak. We first use the Galerkin method to obtain a finite-dimensional ordinary differential equation (ODE) model based on the original magnetic diffusion PDE. Then, we combine the control parameterization method with a novel time-scaling transformation to obtain an approximate optimal parameter selection problem, which can be solved using gradient-based optimization techniques such as sequential quadratic programming (SQP). This control parameterization approach involves approximating the tokamak input signals by piecewise-linear functions whose slopes and break-points are decision variables to be optimized. We show that the gradient of the objective function with respect to the decision variables can be computed by solving an auxiliary dynamic system governing the state sensitivity matrix. Finally, we conclude the paper with simulation results for an example problem based on experimental data from the DIII-D tokamak in San Diego, California.

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Section: Solving Problemmentioning

confidence: 99%

Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

Ren

Lin

et al. 2016

Communications in Nonlinear Science and Numerical Simulation

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show abstract

“…It is further reduced, by the control parametrization technique [22], to a deterministic finite-dimensional problem. A numerical solution of this deterministic optimization problem can be obtained using existing optimization software packages, such as NLPQLP ( [18][19][20]). …”

Section: U Admissiblementioning

confidence: 99%

“…  2 P can be solved as an optimization problem, and various optimization software packages, such as NLPQLP (see [18][19][20]), can be used for this purpose.…”

Section: Existence Of Optimal Solutionsmentioning

confidence: 99%

“…In solving the deterministic constrained optimization problem, we use the control parametrization method (see [18][19][20] …”

Section: ) From Ekeland and Temam [23] A Such That mentioning

confidence: 99%

“…The exponential utility function is the only utility function under which the principle of "zero utility" gives a fair premium that is independent of the level of reserve of an insurance company (see Gerber [17], page 68). In Section 3, by decomposition, we show that this constrained stochastic optimal control problem can be simplified significantly, reducing to a constrained deterministic optimal control problem and then to a finite-dimensional optimization problem, which can be tackled by classical computation techniques, such as [18][19][20]. In Section 4, we demonstrate the effectiveness of this method through carrying out numerical experiments for cases of complete and incomplete markets, where the risky assets are fewer than the uncertainty factors (i.e., the dimension of the Brownian motion).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Optimal Investment and Proportional Reinsurance with Risk Constraint

Liu¹,

Yiu²,

Loxton³

et al. 2013

JMF

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In this paper, we investigate the problem of maximizing the expected exponential utility for an insurer. In the problem setting, the insurer can invest his/her wealth into the market and he/she can also purchase the proportional reinsurance. To control the risk exposure, we impose a value-at-risk constraint on the portfolio, which results in a constrained stochastic optimal control problem. It is difficult to solve a constrained stochastic optimal control problem by using traditional dynamic programming or Martingale approach. However, for the frequently used exponential utility function, we show that the problem can be simplified significantly using a decomposition approach. The problem is reduced to a deterministic constrained optimal control problem, and then to a finite dimensional optimization problem. To show the effectiveness of the approach proposed, we consider both complete and incomplete markets; the latter arises when the number of risky assets are fewer than the dimension of uncertainty. We also conduct numerical experiments to demonstrate the effect of the risk constraint on the optimal strategy.

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Computational optimal control of the Saint–Venant PDE model using the time‐scaling technique

Chen

2015

Asia-Pacific J Chem Eng

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This paper proposes a new time-scaling approach for computational-optimal control of a distributed parameter system governed by the Saint-Venant partial differential equations (PDEs). We propose the time-scaling approach that can change a uniform time partition to a nonuniform one. We also derive the gradient formulas by using the variational method. Then, the method of lines is applied to compute the Saint-Venant PDEs after implementing the time-scaling transformation and the associate costate PDEs. Finally, we compare the optimization results using the proposed time-scaling approach with the one not using it. The simulation result demonstrates the effectiveness of the proposed time-scaling method.By substituting = 0, we can obtain Asia-Pacific Journal of Chemical Engineering COMPUTATIONAL OPTIMAL CONTROL OF THE SAINT-VENANT PDE MODEL 75In this section, we will apply the proposed computational algorithm to an example to verify the effectiveness of the proposed method in this paper.

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Optimal control problems with multiple characteristic time points in the objective and constraints

Cited by 109 publications

References 11 publications

Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

Dynamic optimization of open-loop input signals for ramp-up current profiles in tokamak plasmas

Optimal Investment and Proportional Reinsurance with Risk Constraint

Computational optimal control of the Saint–Venant PDE model using the time‐scaling technique

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