We consider differential systems in RN driven by a nonlinear nonhomogeneous second order differential operator, a maximal monotone term and a multivalued perturbation F(t,u,u′). For periodic systems ...we prove the existence of extremal trajectories, that is solutions of the system in which F(t,u,u′) is replaced by extF(t,u,u′) (= the extreme points of F(t,u,u′)). For Dirichlet systems we show that the extremal trajectories approximate the solutions of the “convex” problem in the C1(T,RN)-norm (strong relaxation).
Periodic solutions for a class of evolution inclusions Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.; Repovš, Dušan D.
Computers & mathematics with applications (1987),
04/2018, Volume:
75, Issue:
8
Journal Article
Peer reviewed
Open access
We consider a periodic evolution inclusion defined on an evolution triple of spaces. The inclusion involves also a subdifferential term. We prove existence theorems for both the convex and the ...nonconvex problem, and we also produce extremal trajectories. Moreover, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (strong relaxation). We illustrate our results by examining a nonlinear parabolic control system.
Consider a continuous-time linear switched system on Rn associated with a compact convex set of matrices. When it is irreducible and its largest Lyapunov exponent is zero there always exists a ...Barabanov norm associated with the system. This paper deals with two types of issues: (a) properties of Barabanov norms such as uniqueness up to homogeneity and strict convexity; (b) asymptotic behavior of the extremal solutions of the linear switched system. Regarding Issue (a), we provide partial answers and propose four related open problems. As for Issue (b), we establish, when n=3, a Poincaré–Bendixson theorem under a regularity assumption on the set of matrices. We then revisit a noteworthy result of N.E. Barabanov describing the asymptotic behavior of linear switched system on R3 associated with a pair of Hurwitz matrices {A,A+bcT}. After pointing out a gap in Barabanov's proof we partially recover his result by alternative arguments.
We consider a time-optimal problem for a car model that can move forward on a plane and turn with a given minimum turning radius. Trajectories of this system are applicable in image processing for ...the detection of salient lines. We prove the controllability and existence of optimal trajectories. Applying the necessary optimality condition given by the Pontryagin maximum principle, we derive a Hamiltonian system for the extremals. We provide qualitative analysis of the Hamiltonian system and obtain explicit expressions for the extremal controls and trajectories.
We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and ...turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in SE2 and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part.
Nonlinear Multivalued Periodic Systems Gasiński, Leszek; Papageorgiou, Nikolaos S.
Journal of dynamical and control systems,
15/4, Volume:
25, Issue:
2
Journal Article
Peer reviewed
Open access
We consider a first-order periodic system involving a time-dependent maximal monotone map, a subdifferential term, and a multivalued perturbation
F
(
t
,
x
). We prove existence theorems for the ...“convex” problem (that is,
F
is convex valued and for the “nonconvex” problem (that is,
F
is nonconvex valued). Also, we establish the existence of extremal trajectories (that is, solutions when the multivalued perturbation
F
(
t
,
x
) is replaced by ext
F
(
t
,
x
), the extreme points of
F
(
t
,
x
)). Also, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (“strong relaxation” theorem). Finally, we illustrate our result by examining a nonlinear periodic feedback control system.
We consider an anti-periodic evolution inclusion defined on an evolution triple of spaces, driven by an operator of monotone-type and with a multivalued reaction term
F
(
t
,
x
). We prove existence ...theorem for the “convex” problem (that is,
F
is convex-valued) and for the “nonconvex” problem (that is,
F
is nonconvex-valued) and we also show the existence of extremal trajectories (that is, when
F
is replaced by
ext
F
). Finally, we prove a “strong relaxation” theorem, showing that the extremal trajectories are dense in the set of solutions of the convex problems.
We consider the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant ...vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin maximum principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates, the vertical and the horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by the Jacobi elliptic functions. Qualitative analysis reveals that the projections of normal extremal trajectories on the
xy
-plane have cusps and inflection points. The vertical subsystem being a generalized pendulum admits reflection symmetries that are used to obtain a characterization of the Maxwell strata.
We formulate a time-optimal problem for a differential drive robot with bounded positive velocities of the driving wheels. This problem is equivalent to a generalization of the classical Markov – ...Dubins problem with an extended domain of control. We classify all extremal controls via the Pontryagin maximum principle. Some optimality conditions are obtained; therefore, the optimal synthesis is reduced to the enumeration of a finite number of possible solutions.
Solutions of the Hamilton–Jacobi equation
H
(
x
,−
Du
(
x
)) = 1, where
H
(·,
p
) is Hölder continuous and the level-sets {
H
(
x
, ·) ≤ 1} are convex and satisfy positive lower and upper curvature ...bounds, are shown to be locally semiconcave with a power-like modulus. An essential step of the proof is the
-regularity of the extremal trajectories associated with the multifunction generated by
D
p
H
.
PDF
