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Person# Milan Korda

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Nonlinear system

In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interes

Problem solving

Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an a

Optimal control

Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has nu

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Colin Neil Jones, Milan Korda, Georgios Stathopoulos

This work presents an algorithmic scheme for solving the infinite-time constrained linear quadratic regulation problem. We employ an accelerated version of a popular proximal gradient scheme, commonly known as the Forward-Backward Splitting (FBS), and prove its convergence to the optimal solution in our infinite-dimensional setting. Each iteration of the algorithm requires only finite memory, is computationally cheap, and makes no use of terminal invariant sets; hence, the algorithm can be applied to systems of very large dimensions. The acceleration brings in ‘optimal’ convergence rates O(1/k2) for function values and O(1/k) for primal iterates and renders the proposed method a practical alternative to model predictive control schemes for setpoint tracking. In addition, for the case when the true system is subject to disturbances or modelling errors, we propose an efficient warm-starting procedure, which significantly reduces the number of iterations when the algorithm is applied in closed-loop. Numerical examples demonstrate the approach.

We study the convergence rate of the moment-sum-of-squares hierarchy of semidefinite programs for optimal control problems with polynomial data. It is known that this hierarchy generates polynomial under-approximations to the value function of the optimal control problem and that these under approximations converge in the L-1 norm to the value function as their degree d tends to infinity. We show that the rate of this convergence is 0(1/log log d). We treat in detail the continuous-time infinite-horizon-discounted problem and describe in brief how the same rate can be obtained for the finite-horizon continuous-time problem and for the discrete-time counterparts of both problems. (C) 2016 Elsevier B.V. All rights reserved.

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This paper investigates the relations between three different properties, which are of importance in optimal control problems: dissipativity of the underlying dynamics with respect to a specific supply rate, optimal operation at steady state, and the turnpike property. We show in a continuous-time setting that if along optimal trajectories a strict dissipation inequality is satisfied, then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state. Finally, we establish novel converse turnpike results, i.e., we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state. We draw upon a numerical example to illustrate our findings. (C) 2017 Elsevier Ltd. All rights reserved.