Pontryagin's maximum principle and sufficient conditions for optimality in the L_0-metric

Consider the following optimal control problem:newline minimize Jleft(uright)=int_{t_{2}}^{t_{1}}f^{0}left(x,u,tright) dt+K_{0}left(x_{1}left(t_{1}right),x_{2}left(t_{2}right)right) subject to overset{cdot}to{x}=fleft(x,u,tright), tinleft[t_1,t_2right], K_1left(x_1left(t_1right),x_2left(t_2right)right)leq0, K_2left(x_1left(t_1right),x_2left(t_2right)right)=0 where u(cdot) is an admissible control and x(cdot) is an absolutely continuous trajectory. If the set of admissible controls is of the form U=left{uleft(cdotright)in L^{infty}left[t_1,t_2right] colon uleft(tright)in Uleft(tright)right} and the functional J(u) has the global minimum J(u^{ast}) in the set U, then u^{ast}(cdot) satisfies Pontryagin's maximum principle. par Let L_{0} denote the space of all measurable functions ucolon[t_1,t_2]rightarrowbold R^n with the metric rholeft(u_1,u_2right)={rm meas}left{tinleft[t_1,t_2right]colon uleft(t_1right)neq uleft(t_2right)right} and let L_0left(Uright) denote the set of all functions uin L_0 such that u(t)in U(t) where U(cdot) is a measurable set-valued mapping. par In the paper the author considers problem (1)--(2) in the special set of admissible controls U_0=L_0(U)cap L^{infty}[t_1,t_2] and proves that each local minimizer u^{ast}in U satisfies Pontryagin's maximum principle. Under additional assumptions the maximum principle is also a sufficient condition for u^{ast}in U_0 to be a local minimizer for problem (1)--(2).