Automatic solution of optimal-control problems. IV. Gradient method (Q760177)

[For part I see ibid. 14, 131-148 (1984; reviewed above).] - The authors' table method for automatic calculations of derivatives of real functions of several variables is used to approximate by the gradient method the solution of the two point boundary value problem: \((x',p')=(H_ p(t,x,p,u),\quad -H_ x(t,x,p,u)),\) \(x(0)=x_ 0\), \(p(T)=0\), \(H_ u(t,x,p,u)=0\) associated by Pontryagin's maximum principle to the problem of minimizing \(\int^{T}_{0}F(t,x(t),u(t))dt\) subject to: \(x'(t)=f(t,x(t),u(t))\), \(x(0)=x_ 0\), x(t),u(t)\(\in R\); the Hamiltonian is defined as usual by: \(H(t,x,p,u)=F(t,x,u)+p.f(t,x,u).\) Numerical experiments for the problem defined by: \(F(t,x,u)=x^ 2+u^ 2\), \(f(t,x,u)=-ax-bx^ 2+u\), \(T=1\), \(x(0)=1\), in the cases: \((a,b)=(1,0.1)\), \((a,b)=(0,1)\) and \((a,b)=(0,0)\) are presented and an offer to provide on request the FORTRAN program listing is made.