Optimal Planar Interception with Fixed End Conditions: Approximate Solutions

In a previous paper (Ref. 1), an exact solution of the optimal planar interception with fixed end conditions was derived in closed form. The optimal control was expressed as an explicit function of the state variables and two fixed parameters, obtained by solving a set of nonlinear algebraic equations involving elliptic integrals. In order to facilitate the optimal control implementation, the present paper derives a highly accurate simplified solution assuming that the ratio of the pursuer turning radius to the initial range is small. An asymptotic expansion further reduces the computational workload. Construction of a near-optimal open-loop control, based on the approximations, completes the present paper.     

Asymptotic solution of a singularly perturbed set of functional-differential equations of Riccati type encountered in the optimal control theory

The boundary-value problem for the set of functional-differential equations with partial derivatives of Riccati type, associated with a singularly perturbed linear-quadratic optimal control problem with delay in state, is considered. The expression for a solution of the problem, which transforms it to the explicit singular perturbation form, is proposed. An asymptotic solution of this problem is constructed. Received August 7, 1997     

Infinite horizon quadratic control of linear singularly perturbed systems with small state delays: an asymptotic solution of Riccati-type equations

Email: valery{at}techunix.technion.ac.il Received on January 31, 2006; Accepted on October 5, 2006 An infinite horizon linear-quadratic optimal control problemfor a singularly perturbed system with multiple point-wise anddistributed small delays in the state variable is considered.The set of Riccati-type equations, associated with this problemby the control optimality conditions, is studied. Since thesystem in the control problem is singularly perturbed, the equationsof this set are also perturbed by a small parameter of the singularperturbations. The zero-order asymptotic solution to this setof equations is constructed and justified. Based on this asymptoticsolution, parameter-free sufficient conditions for the existenceand uniqueness of solution to the original optimal control problemare established.     

Homicidal Chauffeur Game with Target Set in the Shape of a Circular Angular Sector: Conditions for Existence of a Closed Barrier

A planar constant-speed pursuit-evasion problem with dynamic model similar to the one of the homicidal chauffeur game and with prescribed angular constraints in the capture criterion is analyzed as a differential game of kind. Because of the angular constraints, the target set of the game has the shape of a circular angular sector. Conditions for the existence of the game barrier (closed) are obtained. Using these conditions, a necessary and sufficient condition for capture of a slower evader from any initial state of the game is established. This condition is represented by an expression for the minimal nondimensional capture radius, normalized by the pursuer minimal turning radius, which guarantees capture of all slower evaders. This minimal capture radius depends on the angular constraints. Capture from any initial state implies that the barrier of the game does not exist and vice versa. In this game, two types of barrier are derived, with termination at either points of smoothness or points of nonsmoothness (corner points) of the boundary of the target set. The results are illustrated by numerical examples.     

Robust controllability of linear systems: Analysis and application to pursuit problems

Robust controllability conditions for a linear uncertain system are derived. A class of transferring strategies is obtained. A robust controllability set, respecting a given control constraint, is constructed. An application example is presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust pursuit of a hybrid evader

A pursuit-evasion differential game with bounded controls and prescribed duration is considered. The evader has two possible dynamics, while the pursuer dynamics is fixed. The evader can change the dynamics once during the game. The pursuer knows the possible dynamics of the evader, but not the actual one. The optimal pursuer strategy in this game is obtained. It is robust with respect to the control of the evader, the order of its dynamics and the time of the mode change. The capture conditions of the game are established and the pursuer capture zone is constructed. An illustrative example of the game is also presented.     

Optimal Evasion from a Pursuer with Delayed Information

A class of prescribed duration pursuit–evasion problems with first-order acceleration dynamics and bounded controls is considered. In this class, the pursuer has delayed information on the lateral acceleration of the evader, but knows perfectly the other state variables. Moreover, the pursuer applies a strategy derived from the perfect information pursuit–evasion game solution. Assuming that the evader has perfect information on all the state variables as well as on the delay of the pursuer and its strategy, an optimal evasion problem is formulated. The necessary optimality conditions indicate that the evader optimal control has a bang–bang structure. Based on this result, two particular cases of the pursuer strategy (continuous and piecewise continuous in the state variables) are considered for the solution of the optimal evasion problem. In the case of the continuous pursuer strategy, the switch point of the optimal control can be obtained as a root of the switch function. However, in the case of the piecewise continuous (bang–bang) pursuer strategy, this method fails, because of the discontinuity of the switch function at this very point. In this case, a direct method for obtaining the switch point, based on the structure of the solution, is proposed. Numerical results illustrating the theoretical analysis are presented leading to a comparison of the two cases.     

Optimal planar interception with fixed end conditions: Closed-form solution

A planar constant-speed interception with prescribed end conditions is analyzed. The performance index is the time of capture penalized by the control energy. For this problem, the optimal control of the pursuer is obtained in closed form, based on solving a set of nonlinear algebraic equations involving elliptic integrals. The construction of the solution is inspired by the singularly perturbed structure of the nondimensional equations of motion.This research was supported in part by the Ministry of Immigrant Absorption, Center for Absorption in Science, Jerusalem, Israel.     

Radial characteristics of the stimulated emission of copper-vapor lasers

Copper-vapor lasers are currently under investigation with the definite aim of increasing the efficiency and mean power of such lasers. One way of doing this is to increase the repetition frequency of the pulse discharge and increase the volume of the active medium. However, if the interval between the pulses is reduced, cumulative effects begin to play a part in determining the state of the active medium before a new pulse. The main such effects are evidently the accumulation of heat, electrons, and metastable atoms. Ultimately, they reduce the pulse power and mean power of the laser. The reduction is characterized by the fact that the power of the laser emission decreases primarily in the center of the tube, and the transverse structure of the beam acquires an annular shape. Moreover, with increasing diameter of the working channel, these features are manifested at lower repetition frequencies. [1]. The present paper gives experimental data on this phenomenon and an attempt to explain it on the basis of the radial characteristics of the active medium.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 38–43, October, 1980.     

Control synthesis for a nonlinear nonstationary process

A system of nonlinear differential equations is considered. The relationship of the general solution of this system with the general solution of a certain linear system of differential equations is studied. The result leads to the solution of a control problem that produces a given optimal control synthesis.Kommunarnyi Mining and Metallurgical Institute. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, 123–129, 1989.