relative minimum, 243–250 Inception, admissible set of, 255 for optimal control problem, 283–284 Indicatrix, 228 concavity of, 232 convexity of, 232 example of, 232–236 Infinitesimal transformations, 122, 238 Integral constraints, 326 Invariance of Euler-Lagrange equations, 112 Invariant integral, 139 for homogeneous problem, 241 Invariant integral: for Lagrange problem, 374 for problem: in n unknown functions, 168 in two independent variables, 192–193 for variable endpoint

lineal elements we have (See Lemmas 5.1.1 and 5.1.2.) Considering now the characteristic surface at a point of an optimal trajectory and the plane we come to realize, as in Sec. 5.3,that this plane has to be tangent to the characteristic surface for where is the slope of the optimal trajectory at the point We obtain that, by necessity, for every lineal element of an optimal trajectory except where has a jump discontinuity. As before, is to be viewed as a function of x and , being the

y = 3x and sin x, respectively, in Probs. 6.3.2 and 6.3.3 are extremals of the variational problems that are obtained from the stated Lagrange problems by omitting the constraining equations. 7. Solve the Lagrange problem Check whether or not the extremal is singular. *8. Show: If (or < 0) for all when taken for the solutions of the Mayer equations and the constraining equations of the Lagrange problem, and if then 9. Find an extremal of and show that it is anormal. 6.5 THE ISOPERIMETRIC

The existence of ||f|| follows from That (1.4.4) satisfies the requirements of a norm follows from an argument similar to that used in example E. (See Prob. 1.4.9.) A norm as defined in (1.4.2), (1.4.3), or (1.4.4) is usually referred to as a maximum norm or a Chebychev norm. G. and are the spaces Cs1[0,1] and Csp1[0,1], respectively (see example D), in which the norm is defined as follows: We observe that exists for all or (see Prob. 1.4.12) and that it satisfies all the requirements of a

necessary that y0 satisfy the conditions N1 and N3 and N4 and N5. For the extremal to yield a weak relative minimum (maximum) for it is necessary that y0 satisfy the conditions N1 and N2. For the regular extremal to yield a weak relative minimum (maximum) for I[y], it is necessary that y0 satisfy the conditions N1 and N3 and N5. N1 Euler-Lagrange equation (Corollary 2 to Theorem 2.3) N2 Legendre condition (Theorem 7.2) N3 Strengthened Legendre condition (7.3.1) N4 Weierstrass’ necessary