Linear-Quadratic Problems of Optimal Control in the Space of Probabilities

Staritsyn, Maxim; Pogodaev, Nikolay; Pereira, Fernando Lobo

doi:10.1109/lcsys.2022.3184257

Cited by 6 publications

(5 citation statements)

References 20 publications

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“…Furthermore -due to the nonlocal nature of the underlying increment formula -our algorithm can step over local solutions, and therefore, has the potential of global search. 1 This paper generalizes our recent works [12], [13], where the exact increment formula and a nonlocal algorithm were derived for models of linear and linear-quadratic structure. Now, we consider an arbitrary nonlinear cost functional on the space of probability measures, which has the so-called intrinsic derivative (see [14] and the discussion in sec.…”

Section: B Contribution and Noveltysupporting

confidence: 54%

“…Statistical Tracking: In some cases [13], [17], the previous performance criterion could be too rigid. Instead of matching the desired profile in average, one may require that the target distribution has prescribed statistical characteristics, for instance, its expectation and variance approach some desired values.…”

Section: B Problem Specificationmentioning

confidence: 99%

“…On the conceptual level, an iteration of the announced iterative method consists of just three steps: i) integration of the ODE (12) together with the linearized system (14), for ū = u k and various initial conditions over some mesh π M spt ϑ = {y k } M k=0 ⊆ spt ϑ, to obtain (X k , J k ), ii) numeric solution of the PDE ( 4), (6) backfed by (19) with ( X, J) = (X k , J k ), to obtain µ k+1 , and iii) control update u k+1 := w t [µ k+1 t ]. Arguments similar to [13,Appendix B] show that this iterative method converges in the residual of Pontryagin's maximum principle [11] for the convexified problem (P ) as max…”

Section: B Numeric Algorithmmentioning

confidence: 99%

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Optimal Route Planning in Steady Planar Convective Flows

Chertovskih¹,

Staritsyn²,

Pereira³

2020

Lecture Notes in Electrical Engineering

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Motivated by the problem of designing robust composite pulses for Bloch equations in the presence of natural perturbations, we study an abstract optimal ensemble control problem in a probabilistic setting with a general nonlinear performance criterion. The model under study addresses meanfield dynamics described by a linear continuity equation in the space of probability measures. For the resulting optimization problem, we derive an exact representation of the increment of the cost functional in terms of the flow of the driving vector field. Relying on the exact increment formula, a descent method is designed that is free of any internal line search. The numerical method is applied to solve new control problems for distributed ensembles of Bloch equations. *The authors acknowledge the financial support of the Foundation for Science and Technology (FCT, Portugal) in the framework of the Associated Laboratory "Advanced Production and Intelligent Systems" (AL ARISE, ref. LA/P/0112/2020), R&D Unit SYSTEC (base UIDB/00147/2020 and programmatic UIDP/00147/2020 funds), and projects SNAP (ref. NORTE-01-0145-FEDER-000085) and MLDLCOV (ref. DSAIPA/CS/0086/2020).

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Section: B Contribution and Noveltysupporting

confidence: 54%

Section: B Problem Specificationmentioning

confidence: 99%

Section: B Numeric Algorithmmentioning

confidence: 99%

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Optimal Route Planning in Steady Planar Convective Flows

Chertovskih¹,

Staritsyn²,

Pereira³

2020

Lecture Notes in Electrical Engineering

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“…In future research, we are planning to extend the BCD scheme to solve other 3D partial differential equations, such as the 3D Helmholtz equation, 3D parabolic equations, 3D incompressible Navier-Stokes equations, etc. These equations have various applications in science and engineering, such as image reconstruction by hypoelliptic diffusion [37], mathematical neuroscience [38], etc.…”

Section: Discussionmentioning

confidence: 99%

Higher-Order Blended Compact Difference Scheme on Nonuniform Grids for the 3D Steady Convection-Diffusion Equation

Lan

et al. 2023

Axioms

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This paper proposes a higher-order blended compact difference (BCD) scheme on nonuniform grids for solving the three-dimensional (3D) convection–diffusion equation with variable coefficients. The BCD scheme has fifth- to sixth-order accuracy and considers the first and second derivatives of the unknown function as unknowns as well. Unlike other schemes that require grid transformation, the BCD scheme does not require any grid transformation and is simple and flexible in grid subdivisions. Concurrently, the corresponding high-order boundary schemes of the first and second derivatives have also been constructed. We tested the BCD scheme on three problems that involve convection-dominated and boundary-layer features. The numerical results show that the BCD scheme has good adaptability and high resolution on nonuniform grids. It outperforms the BCD scheme on uniform grids and the high-order compact scheme on nonuniform grids in the literature in terms of accuracy and resolution.

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“…An iterative implementation of this idea is outlined in Algorithm 2. The convergence analysis can be elaborated similarly to [23,Appendix B].…”

Section: B Open-loop Controlsmentioning

confidence: 99%