103 research outputs found
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
On the relationship between stochastic turnpike and dissipativity notions
In this paper, we introduce and study different dissipativity notions and
different turnpike properties for discrete-time stochastic nonlinear optimal
control problems. The proposed stochastic dissipativity notions extend the
classic notion of Jan C. Willems to random variables and to probability
measures. Our stochastic turnpike properties range from a formulation for
random variables via turnpike phenomena in probability and in probability
measures to the turnpike property for the moments. Moreover, we investigate how
different metrics (such as Wasserstein or L\'evy-Prokhorov) can be leveraged in
the analysis. Our results are built upon stationarity concepts in distribution
and in random variables and on the formulation of the stochastic optimal
control problem as a finite-horizon Markov decision process. We investigate how
the proposed dissipativity notions connect to the various stochastic turnpike
properties and we work out the link between these two different forms of
dissipativity
An exponential turnpike theorem for dissipative discrete time optimal control problems
revised 2013, 23 p.International audienceWe investigate the exponential turnpike property for nite horizon undercounted discrete time optimal control problems without any terminal constraints. Considering a class of strictly dissipative systems we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property. Two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration
Pathwise turnpike and dissipativity results for discrete-time stochastic linear-quadratic optimal control problems
We investigate pathwise turnpike behavior of discrete-time stochastic
linear-quadratic optimal control problems. Our analysis is based on a novel
strict dissipativity notion for such problems, in which a stationary stochastic
process replaces the optimal steady state of the deterministic setting. The
analytical findings are illustrated by a numerical example
Turnpike and dissipativity in generalized discrete-time stochastic linear-quadratic optimal control
We investigate different turnpike phenomena of generalized discrete-time
stochastic linear-quadratic optimal control problems. Our analysis is based on
a novel strict dissipativity notion for such problems, in which a stationary
stochastic process replaces the optimal steady state of the deterministic
setting. We show that from this time-varying dissipativity notion, we can
conclude turnpike behaviors concerning different objects like distributions,
moments, or sample paths of the stochastic system and that the distributions of
the stationary pair can be characterized by a stationary optimization problem.
The analytical findings are illustrated by numerical simulations
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