On the Geometry of Reachable Sets for Control Systems with Isoperimetric Constraints

Gusev, M. I.; Zykov, I. V.

doi:10.1134/s0081543819020093

Cited by 2 publications

(3 citation statements)

References 9 publications

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“…Using the previous scheme, we can also write this condition in the form of Pontryagin's maximum principle [16] (see also [11]).…”

Section: Multiple Constraints On the Controlmentioning

confidence: 99%

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Computing the Reachable Set Boundary for an Abstract Control System: Revisited

Gusev

2023

UMJ

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A control system can be treated as a mapping that maps a control to a trajectory (output) of the system. From this point of view, the reachable set, which consists of the ends of all trajectories at a given time, can be considered an image of the set of admissible controls into the state space under a nonlinear mapping. The paper discusses some properties of such abstract reachable sets. The principal attention is paid to the description of the set boundary.

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“…Using the previous scheme, we can also write this condition in the form of Pontryagin's maximum principle [16] (see also [11]).…”

Section: Multiple Constraints On the Controlmentioning

confidence: 99%

“…Therefore, this control satisfies the maximum principle. This result was generalized in [11] for several mixed integral constraints in which the integrands depend on both control and state variables. In [9] (see, also [1]), we proposed to consider the reachability problem in terms of nonlinear mappings of Banach spaces.…”

Section: Introductionmentioning

confidence: 98%

Computing the Reachable Set Boundary for an Abstract Control System: Revisited

Gusev

2023

UMJ

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“…Different topological properties and approximate construction methods of the set of trajectories described by various types of the integral and differential equations, where the control functions have integral constraints, are considered in papers [4-8, 11, 13, 14]. In papers [4,5,11,14] the compactness, closedness, path-connectedness properties and approximate construction methods of the set of trajectories and attainable sets of the control systems which are affine with respect to the control vector are discussed. In papers [6][7][8]13] the same problems are investigated for nonlinear control systems.…”

Section: Introductionmentioning

confidence: 99%

On the Properties of the Set of Trajectories of the Nonlinear Control System With Quadratic Integral Constraint on the Control Functions

Hüseyin

2023

UMJ

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In this paper the control system described by a nonlinear differential equation is studied. It is assumed that the control functions have a quadratic integral constraint, more precisely, the admissible control functions are chosen from the ellipsoid of the space \(L_2([t_0,\theta];\mathbb{R}^m)\). Different properties of the set of trajectories are investigated. It is proved that a small perturbation of the set of control functions causes also appropriate small perturbation of the set of trajectories. It is also shown that the set of trajectories has a small change if along with the integral constraint on the control functions, a sufficiently large norm type geometric constraint on the control functions is introduced. It is established that every trajectory is robust with respect to the fast consumption of the remaining control resource, and hence every trajectory of the system can be approximated by a trajectory generated by full consumption of the total control resource.

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On the Geometry of Reachable Sets for Control Systems with Isoperimetric Constraints

Cited by 2 publications

References 9 publications

Computing the Reachable Set Boundary for an Abstract Control System: Revisited

Computing the Reachable Set Boundary for an Abstract Control System: Revisited

On the Properties of the Set of Trajectories of the Nonlinear Control System With Quadratic Integral Constraint on the Control Functions

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