scholarly journals THE OPTIMAL CONTROL PROBLEM FOR ONE-DIMENSIONAL NONLINEAR SHRODINGER EQUATIONS WITH A SPECIAL GRADIENT TERM

2019 ◽  
pp. 49-66
Author(s):  
G. Yagub ◽  
N. S. Ibrahimov ◽  
M. Zengin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Quality Criterion ◽  
Necessary Condition ◽  
Gradient Term ◽  
Complex Coefficient ◽  
Existence And Uniqueness Theorem ◽  
One Dimensional ◽  
Nonlinear Part

In this paper we consider the optimal control problem for a one-dimensional nonlinear Schrodinger equation with a special gradient term and with a complex coefficient in the nonlinear part, when the quality criterion is a final functional and the controls are quadratically summable functions. In this case, the questions of the correctness of the formulation and the necessary condition for solving the optimal control problem under consideration are investigated. The existence and uniqueness theorem for the solution is proved and a necessary condition is established in the form of a variational inequality. Along with these, a formula is found for the gradient of the considered quality criterion.

scholarly journals The Time-Optimal Control Problem of a Kind of Petrowsky System

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 311
Author(s):  
Dongsheng Luo ◽  
Wei Wei ◽  
Hongyong Deng ◽  
Yumei Liao
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Null Controllability ◽  
Time Optimal Control ◽  
Necessary Condition ◽  
Vibration System ◽  
Time Optimal ◽  
Time Optimal Control Problem

In this paper, we consider the time-optimal control problem about a kind of Petrowsky system and its bang-bang property. To solve this problem, we first construct another control problem, whose null controllability is equivalent to the controllability of the time-optimal control problem of the Petrowsky system, and give the necessary condition for the null controllability. Then we show the existence of time-optimal control of the Petrowsky system through minimum sequences, for the null controllability of the constructed control problem is equivalent to the controllability of the time-optimal control of the Petrowsky system. At last, with the null controllability, we obtain the bang-bang property of the time-optimal control of the Petrowsky system by contradiction, moreover, we know the time-optimal control acts on one subset of the boundary of the vibration system.

scholarly journals An optimal control problem involving a class of linear time-lag systems

1986 ◽  
Vol 28 (1) ◽  
pp. 93-113 ◽  
Author(s):  
K. L. Teo ◽  
K. H. Wong ◽  
Z. S. Wu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Control Problems ◽  
Linear Time ◽  
Convergence Property ◽  
Time Lag ◽  
Necessary Condition ◽  
Terminal Constraints ◽  
Linear Control Constraints

A class of convex optimal control problems involving linear hereditary systems with linear control constraints and nonlinear terminal constraints is considered. A result on the existence of an optimal control is proved and a necessary condition for optimality is given. An iterative algorithm is presented for solving the optimal control problem under consideration. The convergence property of the algorithm is also investigated. To test the algorithm, an example is solved.

Optimal Control of a Chaotic Map: Fixed Point Stabilization and Attractor Confinement

1997 ◽  
Vol 07 (02) ◽  
pp. 437-446 ◽  
Author(s):  
C. Piccardi ◽  
L. L. Ghezzi
Keyword(s):  
Optimal Control ◽  
Fixed Point ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Chaotic Attractor ◽  
Chaotic Map ◽  
Optimal Controller ◽  
Control Of Chaos ◽  
One Dimensional ◽  
Suppression Of Chaos

Optimal control is applied to a chaotic system. Reference is made to a well-known one-dimensional map. Firstly, attention is devoted to the stabilization of a fixed point. An optimal controller is obtained and compared with other controllers which are popular in the control of chaos. Secondly, allowance is made for uncertainty and emphasis is placed on the reduction rather than the suppression of chaos. The aim becomes that of confining a chaotic attractor within a prescribed region of the state space. A controller fulfilling this task is obtained as the solution of a min-max optimal control problem.

scholarly journals Optimal control theory with general constraints

1981 ◽  
Vol 23 (2) ◽  
pp. 115-126 ◽  
Author(s):  
John M. Blatt
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Control Theory ◽  
Time Dependent ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
General Constraints ◽  
Elementary Methods ◽  
And Control

AbstractWe consider an optimal control problem with, possibly time-dependent, constraints on state and control variables, jointly. Using only elementary methods, we derive a sufficient condition for optimality. Although phrased in terms reminiscent of the necessary condition of Pontryagin, the sufficient condition is logically independent, as can be shown by a simple example.

Trajectory Planning of Quadrotor Systems for Various Objective Functions

Robotica ◽  
2020 ◽  
Vol 39 (1) ◽  
pp. 137-152
Author(s):  
Hamidreza Heidari ◽  
Martin Saska
Keyword(s):  
Optimal Control ◽  
Path Planning ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Trajectory Planning ◽  
Minimum Principle ◽  
Optimal Path ◽  
Necessary Condition ◽  
Objective Functions ◽  
Wide Range

SUMMARYQuadrotors are unmanned aerial vehicles with many potential applications ranging from mapping to supporting rescue operations. A key feature required for the use of these vehicles under complex conditions is a technique to analytically solve the problem of trajectory planning. Hence, this paper presents a heuristic approach for optimal path planning that the optimization strategy is based on the indirect solution of the open-loop optimal control problem. Firstly, an adequate dynamic system modeling is considered with respect to a configuration of a commercial quadrotor helicopter. The model predicts the effect of the thrust and torques induced by the four propellers on the quadrotor motion. Quadcopter dynamics is described by differential equations that have been derived by using the Newton–Euler method. Then, a path planning algorithm is developed to find the optimal trajectories that meet various objective functions, such as fuel efficiency, and guarantee the flight stability and high-speed operation. Typically, the necessary condition of optimality for a constrained optimal control problem is formulated as a standard form of a two-point boundary-value problem using Pontryagin’s minimum principle. One advantage of the proposed method can solve a wide range of optimal maneuvers for arbitrary initial and final states relevant to every considered cost function. In order to verify the effectiveness of the presented algorithm, several simulation and experiment studies are carried out for finding the optimal path between two points with different objective functions by using MATLAB software. The results clearly show the effect of the proposed approach on the quadrotor systems.

scholarly journals A Note on Optimal Control Problem Governed by Schrödinger Equation

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Yusuf Koçak ◽  
Ercan Çelik ◽  
Nigar Yıldırım Aksoy
Keyword(s):  
Optimal Control ◽  
Variational Inequality ◽  
Schrödinger Equation ◽  
Optimal Control Problem ◽  
Existence And Uniqueness ◽  
Schrodinger Equation ◽  
Optimal Solution ◽  
Necessary Condition ◽  
Existence And Uniqueness Theorem ◽  
The Cost

AbstractIn this work, we present some results showing the controllability of the linear Schrödinger equation with complex potentials. Firstly we investigate the existence and uniqueness theorem for solution of the considered problem. Then we find the gradient of the cost functional with the help of Hamilton-Pontryagin functions. Finally we state a necessary condition in the form of variational inequality for the optimal solution using this gradient.

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