This Research was supported by the Anadolu University Research Foundation.

A numerical method for building the reachable sets of control system is considered in [1, 2]. The behaviour of the system is described by a differential equation

× x   =f(t, x)+B(t, x)u,   x(t0)=x0,

(1)

where x in Rⁿ, u in R^m - is a control vector, f(t, x) - is a n - dimension vector function, B(t, x) - is a n×m - dimension matrix function. Assume that the realization u(t),  t in [t₀, \theta], of control u satisfies the restriction

ó õ \theta t0  ||u(t)||p dt <= \mu*p,    \mu* >= 0,    p > 1.

(2)

Suppose the following conditions are verified.

A) the functions f(t, x) and B(t, x) are continuous on (t, x) and locally Lipschitz on x, i.e. ||f(t, x^*)-f(t, x_*)|| <= L₁(D)||x^*-x_*||,  ||B(t, x^*)-B(t, x_*)|| <= L₂(D)||x^*-x_*|| where D subset T ×Rⁿ is an arbitrary bounded domain, (t, x_i) in D,   L_i(D)=const,   i=1, 2, ||·|| is the Euclidean norm.

B) there exist constants  c_i in [0, \infty), i=1, 2 such that ||f(t, x)|| <= c₁(1+||x ||),   ||B(t, x)|| <= c₂(1+ ||x ||) for every (t, x) in [t₀, \theta] ×Rⁿ.

Every function u(·) in L_p[t₀, \theta], satisfying the inequality (2) is said to be an admissible control function. By the symbol U we denote the set of all admissible control functions u(·). By the symbol X(t₀, x₀) we denote the totality of all solutions of the system (1), generated by admissible control functions u(·) in U. We set

X(t;t₀, x₀)={ x(t) in Rⁿ: x(·) in X(t₀, x₀)},

Z(t₀, x₀)={ (t, x(t)) in [t₀, \theta] ×Rⁿ : x(·) in X(t₀, x₀) }.

X(t;t₀, x₀) is called the reachable set of the control system (1) at the time moment t under restriction (2). It is possible to prove that there exists a bounded domain D subset [t₀, \theta] ×Rⁿ such that Z(t₀, x₀) subset D. Let H in (0, \infty) be a given number, \Gamma = { t₀ < t₁ < ... < t_N-1 < t_N=\theta} be a partition of the segment [t₀, \theta], where \Delta = (t_i+1-t_i)=(\theta-t₀)/N, i=0, 1, ..., N-1, and \Gamma^*={ 0=y₀ < y₁ < ... < y_r=H } be a partition of the segment [0, H] where y_j+1-y_j=[ H/r]=\Delta^*, j=0, 1, ..., r-1. Let S={ u in R^m : ||u|| = 1 } - be a unit sphere of the space R^m, \Lambda = { s₀, s₁, ... , s_k } be a finite \delta - lattice of the sphere S, where \delta > 0 - is a given number. By the symbol Z(\theta;t₀, x₀) we denote the set of points z(t_N), calculated by formula

z(ti+1)=z(ti)+\Delta[f(ti, z(ti))+B(ti, z(ti))y(ji)  s(li)],

z(t0)=x(t0)=x0,   y(ji) in \Gamma*,   y(li) in \Lambda,   i=0, 1, ..., N-1,

where for the scalar variables y(j₀), y(j₁), ... , y(j_N-1) the inequalities

\Delta· N-1 å i=0  y(ji)p <= \mu*p,   0 <= y(ji) <= H,  i=0, 1, ..., N-1

are verified. Let \alpha(·, ·) be the Hausdorff distance. The following theorem is proved.

Theorem. For any \epsilon > 0 there exist H > 0,  \Delta > 0,  \Delta^* > 0,  \delta > 0 such that

\alpha(X(\theta;t0, x0), Z(\theta;t0, x0)) <= \epsilon

References

[1] Guseinov, Kh.G., Neznakhin, A. A. and Ushakov, V. N. Approximate Construction of Attainability Sets of Control Systems with Integral Constraints on Control, J. Appl. Math. Mechs., 1999, 63, 557-567.

[2] Guseinov, Kh. G., Orhan Ozer and Emrah Akyar, On the Properties of the Reachable Sets of the Nonlinear Control Systems with Integral Constraints on Control, Accepted by the 9th Mediterranean Conference on Control and Automation, June 27-29, 2001, Dubrovnik, Croatia.