14 research outputs found
Infinite horizon sparse optimal control
A class of infinite horizon optimal control problems involving -type
cost functionals with is discussed. The existence of optimal
controls is studied for both the convex case with and the nonconvex case
with , and the sparsity structure of the optimal controls promoted by
the -type penalties is analyzed. A dynamic programming approach is
proposed to numerically approximate the corresponding sparse optimal
controllers
Strong local optimality for generalized L1 optimal control problems
In this paper, we analyse control affine optimal control problems with a cost
functional involving the absolute value of the control. The Pontryagin
extremals associated with such systems are given by (possible) concatenations
of bang arcs with singular arcs and with inactivated arcs, that is, arcs where
the control is identically zero. Here we consider Pontryagin extremals given by
a bang-inactive-bang concatenation. We establish sufficient optimality
conditions for such extremals, in terms of some regularity conditions and of
the coercivity of a suitable finite-dimensional second variation.Comment: Journal of Optimization Theory and Applications, Springer Verlag, In
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Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential
his paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn--Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before. The results obtained in this paper also improve the known results on this phase field model for the case without sparsity
Second-order sufficient conditions for sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions
In this paper we study the optimal control of a parabolic initial-boundary
value problem of Allen--Cahn type with dynamic boundary conditions. Phase field
systems of this type govern the evolution of coupled diffuse phase transition
processes with nonconserved order parameters that occur in a container and on
its surface, respectively. It is assumed that the nonlinear function driving
the physical processes within the bulk and on the surface are double well
potentials of logarithmic type whose derivatives become singular at the
boundary of their respective domains of definition. For such systems, optimal
control problems have been studied in the past. We focus here on the situation
when the cost functional of the optimal control problem contains a
nondifferentiable term like the -norm leading to sparsity of optimal
controls. For such cases, we derive second-order sufficient conditions for
locally optimal controls
On the monotone and primal-dual active set schemes for -type problems,
Nonsmooth nonconvex optimization problems involving the quasi-norm,
, of a linear map are considered. A monotonically convergent
scheme for a regularized version of the original problem is developed and
necessary optimality conditions for the original problem in the form of a
complementary system amenable for computation are given. Then an algorithm for
solving the above mentioned necessary optimality conditions is proposed. It is
based on a combination of the monotone scheme and a primal-dual active set
strategy. The performance of the two algorithms is studied by means of a series
of numerical tests in different cases, including optimal control problems,
fracture mechanics and microscopy image reconstruction
Sparse and switching infinite horizon optimal controls with mixed-norm penalizations
© 2020 EDP Sciences, SMAI. A class of infinite horizon optimal control problems involving mixed quasi-norms of Lp-type cost functionals for the controls is discussed. These functionals enhance sparsity and switching properties of the optimal controls. The existence of optimal controls and their structural properties are analyzed on the basis of first order optimality conditions. A dynamic programming approach is used for numerical realization
Sparse approximation in learning via neural ODEs
We consider the neural ODE and optimal control perspective of supervised
learning with control penalties, where rather than
only minimizing a final cost for the state, we integrate this cost over the
entire time horizon. Under natural homogeneity assumptions on the nonlinear
dynamics, we prove that any optimal control (for this cost) is sparse, in the
sense that it vanishes beyond some positive stopping time. We also provide a
polynomial stability estimate for the running cost of the state with respect to
the time horizon. This can be seen as a \emph{turnpike property} result, for
nonsmooth functionals and dynamics, and without any smallness assumptions on
the data, both of which are new in the literature. In practical terms, the
temporal sparsity and stability results could then be used to discard
unnecessary layers in the corresponding residual neural network (ResNet),
without removing relevant information.Comment: 24 pages, 5 figure