NESTED BOUNDS FOR SOLUTIONS OF DIFFERENTIAL EQUATIONS
For a particular class of ordinary differential equations, an iterative procedure is described that gives a sequence of nested pairs of lower and upper bounds for the solution. Simple conditions are found under which the bounds converge to the solution. The method is used to study equations of Lanchester's type, which have applications to the deterministic description of situations in naval warfare. INTRODUCTION The dynamics of combat in general, and situations in naval warfare in particular, can be described deterministically by means of differential equations usually called Lanchester equations, Lanchester being among the first to investigate such descriptions of combat [Ref. 1] For those equations to provide useful mathematical models, it is desirable that information concerning the qualitative behaviour of the solutions can be obtained without too much effort. Indeed, even if particular solutions can always be found by numerical integration, it is of more fundamental interest to be able to predict, for instance, the effect of parameters and initial conditions on the final result of the combat or on the value of the variables at a given time. The qualitative behaviour of the solutions id studied here through the determination of successive lower and upper bounds. The first steps of the iteration, obtained by hand computation, will provide bounds for the variables (which might represent, for instance, force levels) that depend explicitly on the parameters. It is then easy to find, for instance, bounds for the time required for enemy or own forces to fall below a given value. Besides its usefulness in a qualitative analysis, the procedure developed here can sometimes be adapted for implementation on digital computers in order to determine, by rapidly converging approximations, the exact solution. This is justified when the successive linear differential equations can be replaced by successive algebraic equations; an example of such a situation is given below. Although this work has been inspired by the consideration of equations of Lanchester's type, the results are by no means restricted to such equations, as appears clearly from the hypotheses made on the equations. On the other hand, it must be mentioned that this note is not concerned with the elaboration of models. For these, the reader is referred, for example, to B.W. Conolly [Ref. 2] and to K.M. Mjelde and R.R. Wiederkehr [Ref. 3] who analyse situations related to the defence of merchant shipping against submarines. Although the work of K.M. Mjelde and R.R. Wiederkehr is also concerned with the study of the qualitative behaviour of the solutions of Lanchester equations, its emphasis differs somewhat from our own, which gives greater weight to the mathematical aspects of the problem. 1, STATEMENT OF THE PROBLEM AND NOTATIONS Let us consider a system of differential equations in vector notation: i Ax i{t,z) y, Y tE (the n~dimensional euclidian space) A is an nxn matrix, is the derivative of with respect to t. Systems of type [1] will be investigated in this work, with the important restriction that the elements a^ of A will be assumed to be non negative, except those on the main diagonal, which can be of either sign; a^j s 0 (i 7^ j) If a given system does not satisfy those conditions, an appropriate transformation might force it to do so. A class of particularly simple transformations of that kind is considered in Appendix A. Another possibility would be to include in Y(t,x) the linear terms a. y. having an unacceptable sign. However, this will in general make our method less efficient. As vectors in E will often be compared, it is useful to introduce the following notations, for x^Zj^ ^n (1 s i n) (1 i n) (1 s i n) (1 i n) z _z if y+ z 1 z z^ if y+ z 1 z X if ^i X X z X if ^i X X 2. RELATIONS BETWEEN SOLUTIONS OF DIFFERENTIAL EQUATIONS Theorem 1 If a. s 0 (i 7^ j) and if G(t) is any function satisfying the relation I[t,x(t) G(t) for t 0, [2] the solutions of the differential equations i APS Y(t,2) Z(0) =a [3] z Az G(t) z(0) b a [4] are such that y(t) z(t) for t s 0. [5] PROOF, Let us consider the set S^ of values t s Q for which y^(t) z^(t) and let S ^U^ S^ If S is not void, it has an infimum 0 and a. j(fc) z(t) for 0 s t SS b. 3i y^C? Z i^C? Then, YiCl) (A^ Y. (5, (Az) G^ z^ and a positive number 6 exists, such that y^Ct) y^(SS) z^(t) z^(SS) t t I for Ib t I Consequently, y^Ct) z^(t) for SS 6 t SS the contradiction with [6] shows that S must be void. Corollary 1 If the condition b a^ is replaced by the less stringent condition b a, a similar result is obtained, conclusion [5] being replaced by j(t) z(t) for t 0 [7] PROOF. If a. b. for some i(l i^ n) y^(0) (Aa) Y^(0,a) (Ab) G^(0) z^(0) Consequently, for a sufficiently small positive number e. the relation y.(t) z.(t) is valid for 0 t ^S. More generally, a positive number EUR exists such that 2;(t) z(t) for 0 t e Then, by Theorem 1, (t) z(t) for t s e and Corollary 1 is proved. Corollary 2 If condition [2] is replaced by i I[t,y(t) s G(t) for t 0 [8] conclusion [7] must be replaced by (t) z(t) for t s 0 I [9] PROOF. Let e be a monotonically decreasing sequence tending n (n) toward zero. Because of Corollary 1, the solution z/ (t) of the equation (n) A z^" G(t) +e^ h z^^ho)=a where all the components of h are equal to 1, is such that j{t) z* "(t) for t 0 By the continuity of the solution with respect to a parameter, the relation _2: fo^ t s 0 is established for the solution z_(t) of z Az G(t) z(0) a. If lower bounds are considered instead of upper bounds, similar results are obtained. For instance, the following theorem can be proved. Theorem 2 If a. s: 0 (i 7^ j) and if F(t) is any function satisfying the relation F(t) Y[t,y(t) for t s 0 the solutions of the differential equations i APS Y(t,x) z(0) a X Ax F(t) x(0) a are such that x( Z( fo^ t 0