% script_BrysonProblem_AnalyticSolution % thanks to Martin Neuenhofen for providing script t0 = 0; tE = 1.5; t1 = 0.5/tanh(1.5); % t2 is unique solution of % (0.5+t1-0.5*t1^2)*exp(t1-t2)==te-t2+0.5*(tE-t2)^2 % in interval [t0,tE]. t2 = 1.05462; figure; hold on; box on; axis manual; axis([t0,tE,-1.5,1.5]); grid on; % interval [t0,t1] u = @(t) 1+0*t; x2 = @(t) -t; x1 = @(t) 0.5+t-0.5*t.^2; t = linspace(t0,t1,1000); plot(t,x1(t),'blue','linewidth',1); plot(t,x2(t),'red','linewidth',1); plot(t,u(t) ,'black','linewidth',1); % interval [t1,t2] x1_0 = x1(t1); x2_0 = x2(t1); x1 = @(t) x1_0 ./ exp(t-t1); x2 = @(t) x1_0 * sinh(t-t1) + x2_0 * exp(t-t1); u = @(t) -x1(t)-x2(t); t = linspace(t1,t2,1000); plot(t,x1(t),'blue','linewidth',1); plot(t,x2(t),'red','linewidth',1); plot(t,u(t) ,'black','linewidth',1); % interval [t2,tE] u = @(t) -1+0*t; x2 = @(t) -1.5+t; x1 = @(t) (tE-t)+0.5*(tE-t).^2; t = linspace(t2,tE,1000); plot(t,x1(t),'blue','linewidth',1); plot(t,x2(t),'red','linewidth',1); plot(t,u(t) ,'black','linewidth',1); legend({'\$x_1(t)\$','\$x_2(t)\$','\$u(t)\$'},... 'Interpreter','LateX','fontsize',12); title('Bryson Singular Arc Optimal Control Solution',... 'Interpreter','LateX','fontsize',12); xlabel('time \$t\$','Interpreter','LateX','fontsize',12); ylabel('states \$x_1(t),x_2(t)\$\,, control \$u(t)\$',... 'Interpreter','LateX','fontsize',12);