Advances in Pure Mathematics, 2013, 3, 685-688

Published Online November 2013 (http://www.scirp.org/journal/apm)

http://dx.doi.org/10.4236/apm.2013.38092

Open Access APM

Saddle Point Solution for System with

Parameter Uncertainties

Abiola Bankole, T. C. Obiwuru

Department of Ac tuarial Scienc e and Insurance, Faculty of Business Administration, University of Lagos, Lagos, Nigeria

Email: abankole2006@yahoo.com, sirtimmyo@yahoo.com

Received September 26, 2013; revised October 26, 2013; accepted November 3, 2013

Copyright © 2013 Abiola Bankole, T. C. Obiwuru. This is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

In this paper, we consider dynamical system, in the presence of parameter uncertainties. We apply max-min principles

to determine the saddle point solution for the class of differential game arising from the associated dynamical system.

We also provide sufficient condition for the existence of this saddle point.

Keywords: Parameter Uncertainty; Min-Max Principles; Saddle Point; Differential Games

1. Introduction

The central goal of a manager is to seek ways of control-

ling his environment, so that he can have a considerable

degree of influence on the system in which he operates.

He does this for the following reasons:

• He wants to maximize the system to his own benefit.

• He wants the system to remain stable in state, and

does not dri ft to und esi rable steady state.

• He wants to enjoy continually a steady state of maxi-

mum benefit, even in the presence of arbitrary envi-

ronmental disturbances.

A Lot of research work has been devoted to control-

ling uncertainties see e.g . [1 -4].

In dynamical system, three types of uncertainties are

normally encountered na mely:

1) Uncertainty in the model (parameter);

2) Uncertainty in the input (disturbance); and

3) Uncertainty in the state.

This paper deals with the first type of the uncertainty

while the other types have been dealt with by the author

and other researchers in [1,3,4], etc. We also assume here

that the state of the system under consideration is per-

fectly available for measurement.

The classical method of studying perturbation in a

nonlinear system is to approximate its behaviour by lin-

earizing the system in the neighbourhood of a steady

state. Such analysis proves suitable for many systems,

but only for small initial perturbation. In this paper we

are considering systems with parameter uncertainties,

and therefore a different approach is required. We use

zero-sum game approach. We introduce appropriate cost

functional which is required to be minimised by the con-

trol and maximised by the uncertainty. The zero-sum

game allow us for consideration of saddle point solution,

which leads to “Worst case design concept”.

2. Problem Formulation

Consider the following dynamical system in the presence

of parameter uncertainty defined by:

() ()

()

() ()

()

()

,,

tF tvtxtGtvtut

′=+

(1)

()

00 0

,,

txttT=∈ (2)

where

()

()

()

00

,ii

i

tvtF tvF

=

=+

(3)

is matrix.

,nn×

()

0.F is continuous on

0,tT, also i

1.2, ,i=

are constant matrices

nn×

()

12 1

,,,vvvV IR∈⊆

T

()

1

T

112

,,, ;iv

Vvvvv

=

≤

(4)

()

()

()

1

,Gtvtv Gt

+

=

is an matrix. is nn×

()

.G

continuous on

0,tT

1

12 2

,1,vVIRV q

+∈⊆ =

, q is a

given scalar.

()

n

tIR∈ (state vector), (control vector).

()

m

ut IR∈

We shall be interested in determining a stable control

of (1) under some parameter uncertainties

()

.vt