131 research outputs found
Steady-state and periodic exponential turnpike property for optimal control problems in hilbert spaces
First Published in SIAM Journal on Control and Optimization in
Volume 56, Issue 2, 2018, Pages 1222-1252, published by the Society for Industrial and Applied Mathematics (SIAM)In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reects the fact that, in large control time horizons, the optimal state and control and adjoint state remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking termsThe authors acknowledge the nancial support by the grant FA9550-14-1-0214 of the
EOARD-AFOSR. The second author was partially supported by the National Natural Science Foundation of China under grants 11501424 and 11371285. The third author was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, FA9550-15-1-0027 of AFOSR, the MTM2014-52347 and MTM2017-92996 grants of the MINECO (Spain), and ICON of the French AN
Integral and measure-turnpike properties for infinite-dimensional optimal control systems
We first derive a general integral-turnpike property around a set for
infinite-dimensional non-autonomous optimal control problems with any possible
terminal state constraints, under some appropriate assumptions. Roughly
speaking, the integral-turnpike property means that the time average of the
distance from any optimal trajectory to the turnpike set con- verges to zero,
as the time horizon tends to infinity. Then, we establish the measure-turnpike
property for strictly dissipative optimal control systems, with state and
control constraints. The measure-turnpike property, which is slightly stronger
than the integral-turnpike property, means that any optimal (state and control)
solution remains essentially, along the time frame, close to an optimal
solution of an associated static optimal control problem, except along a subset
of times that is of small relative Lebesgue measure as the time horizon is
large. Next, we prove that strict strong duality, which is a classical notion
in optimization, implies strict dissipativity, and measure-turnpike. Finally,
we conclude the paper with several comments and open problems
Turnpilke property for infinite-dimensional generalized LQ problem
We provide a characterization of the exponential (integral) turnpike property
for infinite dimensional generalized linear-quadratic optimal control problems
in terms of structural properties of the control system, such as exponential
stabilizability and detectability. The proof relies on the analysis of the
exponential convergence of solutions to the differential Riccati equations to
the algebraic counterpart, and on a necessary condition for exponential
stabilizability in terms of a closed range test
The exponential turnpike property for periodic linear quadratic optimal control problems in infinite dimension
In this paper, we establish an exponential periodic turnpike property for
linear quadratic optimal control problems governed by periodic systems in
infinite dimension. We show that the optimal trajectory converges exponentially
to a periodic orbit when the time horizon tends to infinity. Similar results
are obtained for the optimal control and adjoint state. Our proof is based on
the large time behavior of solutions of operator differential Riccati equations
with periodic coefficients
Exponential Turnpike property for fractional parabolic equations with non-zero exterior data
We consider averages convergence as the time-horizon goes to infinity of
optimal solutions of time-dependent optimal control problems to optimal
solutions of the corresponding stationary optimal control problems. Control
problems play a key role in engineering, economics and sciences. To be more
precise, in climate sciences, often times, relevant problems are formulated in
long time scales, so that, the problem of possible asymptotic behaviors when
the time-horizon goes to infinity becomes natural. Assuming that the controlled
dynamics under consideration are stabilizable towards a stationary solution,
the following natural question arises: Do time averages of optimal controls and
trajectories converge to the stationary optimal controls and states as the
time-horizon goes to infinity? This question is very closely related to the
so-called turnpike property that shows that, often times, the optimal
trajectory joining two points that are far apart, consists in, departing from
the point of origin, rapidly getting close to the steady-state (the turnpike)
to stay there most of the time, to quit it only very close to the final
destination and time. In the present paper we deal with heat equations with
non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the
fractional Laplace operator (). We prove the turnpike
property for the nonlocal Robin optimal control problem and the exponential
turnpike property for both Dirichlet and nonlocal Robin optimal control
problems
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