Calculus of Variations and Nonlinear Optimization Based Algorithm for Optimal Control of Hybrid Systems with Controlled Switching

Algorithm 1

Step 1. Chose an initial for ;

Step 2. Compute the corresponding continuous control

 (i) Solve the Hamiltonian system using the shooting method

  (a) Guess the unspecified initial conditions

  and ;

  (b) Integrate the Hamiltonian system (33) forward from

  to ;

  (c) Using the resulting values of and , evaluate

  the error function:

    ;

       ;

  (d) Adjust the value of and , using a

  numerical method for solving nonlinear equations, to bring

  the function E closer to zero;

 (ii) Calculate the continuous optimal control (32);

Step 3. Calculate J (16);

Step 4. Calculate (31);

Step 5. Find the value of by using a nonlinear optimization

   technique; ;

Step 6. Repeat Steps , , and until the criterion

    is satisfied.