ISSN:

2155-3289

eISSN:

2155-3297

## Numerical Algebra, Control & Optimization

2013 , Volume 3 , Issue 1

Special Issue dedicated to Professor George Leitmann on the occasion of his 88th birthday (Part I)

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2013, 3(1): i-iii
doi: 10.3934/naco.2013.3.1i

*+*[Abstract](1266)*+*[PDF](1203.9KB)**Abstract:**

This issue of Numerical Algebra, Control and Optimization is dedicated to Professor George Leitmann on the occasion of his 88th birthday. George Leitmann was born in 1925 in Vienna, Austria. He migrated to USA and received his BA and MS in physics in 1949 and 1950 from Columbia University. During the following seven years at the US Naval Ordnance Test Station, China Lake, California, he worked mostly on rocket trajectory optimization and testing. During this period he also completed a PhD in Engineering Science at the University of California at Berkeley. He joined the University of California at Berkeley in 1957, and became Professor of Engineering Science in 1963 until his retirement in 1991. Since then, he has been Professor Emeritus of Engineering Science. He has been Professor in the Graduate School since 1995.

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2013, 3(1): 1-30
doi: 10.3934/naco.2013.3.1

*+*[Abstract](2335)*+*[PDF](2388.4KB)**Abstract:**

In this paper, we address the problem of jamming in a communication network within a team of mobile autonomous agents. In contradistinction with the contemporary research regarding jamming, we model the intrusion as a pursuit-evasion game between a mobile jammer and a team of agents.

First, we consider a differential game-theoretic approach to compute optimal strategies for a team of UAVs trying to evade a jamming attack initiated by an aerial jammer in their vicinity. We formulate the problem as a zero-sum pursuit-evasion game, where the cost function is the termination time of the game. We use

*Isaacs'*approach to obtain necessary conditions to arrive at the equations governing the saddle-point strategies of the players. We illustrate the results through simulations. Next, we analyze the problem of jamming from the perspective of maintaining connectivity in a network of mobile agents in the presence of an adversary. This is a variation of the standard connectivity maintenance problem in which the main issue is to deal with the limitations in communications and sensing model of each agent. In our work, the limitations in communication are due to the presence of a jammer in the vicinity of the mobile agents. We compute evasion strategies for the team of vehicles based on the connectivity of the resultant

*state-dependent graph*. We present some simulations to validate the proposed control scheme. Finally, we address the problem of jamming for the scenario in which each agent computes its control strategy based on limited information available about its neighbors in the network. Under this decentralized information structure, we propose two approximation schemes for the agents and study the performance of the entire team for each scheme.

2013, 3(1): 31-48
doi: 10.3934/naco.2013.3.31

*+*[Abstract](1491)*+*[PDF](2783.1KB)**Abstract:**

In this paper we consider a problem of designing control laws for multiple mobile agents trying to accomplish three objectives. One of the objectives is to sense a given compact domain while satisfying the other objective which is to avoid collisions between the agents themselves as well as with the obstacles. To keep the communication links between the agents reliable, the agents need to stay relatively close during the sensing operation which is the third and final objective. The design of control laws is based on carefully constructed objective functions and on an assumption that the agents' dynamic models are nonlinear yet affine in control laws. As an illustration of some performance characteristics of the proposed control laws, a numerical example is provided.

2013, 3(1): 49-62
doi: 10.3934/naco.2013.3.49

*+*[Abstract](1636)*+*[PDF](336.0KB)**Abstract:**

It is well known that the free motion of a single-degree-of-freedom damped linear dynamical system can be characterized as overdamped, underdamped, or critically damped. Using the methodology of phase synchronization, which transforms any system of linear second-order differential equations into independent second-order equations, this characterization of free motion is generalized to multi-degree-of-freedom damped linear systems. A real scalar function, termed the viscous damping function, is introduced as an extension of the classical damping ratio. It is demonstrated that the free motion of a multi-degree-of-freedom system is characterized by its viscous damping function, and sometimes the characterization may be conducted with ease by examining the extrema of the viscous damping function.

2013, 3(1): 63-76
doi: 10.3934/naco.2013.3.63

*+*[Abstract](1472)*+*[PDF](248.0KB)**Abstract:**

This paper provides a new and surprising reason for growth, namely costs. More precisely, adding adjustment costs of the control to a one-dimensional, strictly concave optimal control problem does not affect the steady state(s). Then, sufficiently high adjustment costs turn an interior and saddle-point stable steady state of the original, one-state variable model into a source that can lead to unbounded growth. Given a version of the open economy Ramsey model, the initial conditions determine whether unbounded growth or impoverishment results. Related to this threshold property, the strict concave two-state variable control model allows for thresholds even if it has a unique and stable steady state.

2013, 3(1): 77-94
doi: 10.3934/naco.2013.3.77

*+*[Abstract](1587)*+*[PDF](401.4KB)**Abstract:**

We introduce and develop the Hahn symmetric quantum calculus with applications to the calculus of variations. Namely, we obtain a necessary optimality condition of Euler--Lagrange type and a sufficient optimality condition for variational problems within the context of Hahn's symmetric calculus. Moreover, we show the effectiveness of Leitmann's direct method when applied to Hahn's symmetric variational calculus. Illustrative examples are provided.

2013, 3(1): 95-108
doi: 10.3934/naco.2013.3.95

*+*[Abstract](1711)*+*[PDF](194.0KB)**Abstract:**

This paper is concerned with open-loop Stackelberg equilibria of two-player linear-quadratic differential games with mixed leadership. We prove that, under some appropriate assumptions on the coefficients, there exists a unique Stackelberg solution to such a differential game. Moreover, by means of the close interrelationship between the Riccati equations and the set of equations satisfied by the optimal open-loop control, we provide sufficient conditions to guarantee the existence and uniqueness of solutions to the associated Riccati equations with mixed-boundary conditions. As a result, the players' open-loop strategies can be represented in terms of the system state.

2013, 3(1): 109-125
doi: 10.3934/naco.2013.3.109

*+*[Abstract](1804)*+*[PDF](188.2KB)**Abstract:**

An iterative algorithm, which is called the integrated optimal control and parameter estimation algorithm, is developed for solving a discrete time nonlinear stochastic control problem. It is based on the integration of the principle of model-reality differences and Kalman filtering theory, where the dynamic integrated system optimization and parameter estimation algorithm are used interactively. In this approach, the weighted least-square output residual is included in the cost function by appropriately monitoring the weighted matrix. An improved linear quadratic Gaussian optimal control model, rather than the original optimal control problem, is solved. Subsequently, the model optimum is updated using the adjusted parameters induced by the differences between the real plant and the model used. These updated solutions converge to the true optimum, despite model-reality differences. For illustration, the optimal control of a nonlinear continuous stirred tank reactor problem is considered and solved by using the method proposed.

2013, 3(1): 127-143
doi: 10.3934/naco.2013.3.127

*+*[Abstract](1877)*+*[PDF](507.7KB)**Abstract:**

Synthesis of trajectory following optimization methods with Lyapunov optimizing control techniques create continuous controls suitable for systems with switching surfaces. Fundamental contributions include the optimization structure from which the control law is derived and mitigation of undesirable system behavior. It is assumed the analyst is concerned with minimization of control effort and a positive definite function of the state. Disturbance rejection and stability are primary control objectives. Given these priorities, a cost functional with optimal control inspired structure due to the use of minimum cost descent control is constructed. It contains an analyst defined state dependent cost function. Another term is included which is a function of the control effort modified appropriately to reflect that sufficient control is needed to drive the state to a switching surface. This derivation establishes optimization rationale for disturbance rejection and switching surface placement. The control effort term provides for mitigation of finite time interval switching control; a continuous time analog to discontinuous chatter. A different perspective on chatter elimination via sensitivity analysis is provided and inspires the final ``control damping" algorithm.

2013, 3(1): 145-160
doi: 10.3934/naco.2013.3.145

*+*[Abstract](1661)*+*[PDF](194.4KB)**Abstract:**

We prove Euler--Lagrange type equations and transversality conditions for generalized infinite horizon problems of the calculus of variations on time scales. Here the Lagrangian depends on the independent variable, an unknown function and its nabla derivative, as well as a nabla indefinite integral that depends on the unknown function.

2013, 3(1): 161-173
doi: 10.3934/naco.2013.3.161

*+*[Abstract](2529)*+*[PDF](843.9KB)**Abstract:**

The purpose of the present paper is to show that the most prominent results in optimal control theory, the distinction between state and control variables, the maximum principle, and the principle of optimality, resp. Bellman's equation are immediate consequences of Carathéodory's achievements published about two decades before optimal control theory saw the light of day.

2013, 3(1): 175-201
doi: 10.3934/naco.2013.3.175

*+*[Abstract](1694)*+*[PDF](327.7KB)**Abstract:**

The concept of

**incremental quadratic stability ($\delta$QS)**is very useful in treating systems with persistently acting inputs. To illustrate, if a time-invariant $\delta$QS system is subject to a constant input or $T$-periodic input then, all its trajectories exponentially converge to a unique constant or $T$-periodic trajectory, respectively. By considering the relationship of $\delta$QS to the usual concept of quadratic stability, we obtain a useful necessary and sufficient condition for $\delta$QS. A main contribution of the paper is to consider nonlinear/uncertain systems whose state dependent nonlinear/uncertain terms satisfy an

**incremental quadratic constraint**which is characterized by a bunch of symmetric matrices we call

**incremental multiplier matrices**. We obtain linear matrix inequalities whose feasibility guarantee $\delta$QS of these systems. Frequency domain characterizations of $\delta$QS are then obtained from these conditions. By characterizing incremental multiplier matrices for many common classes of nonlinearities, we demonstrate the usefulness of our results.

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