need to build the optimal controller, i.e. one that maximizes: J = ∫ 0 t f f ( u ) d tFor the following time-dependent system: x ˙ = g ( x , u , t )where x ( t ) ∈ R is the state, u ( t ) ∈ R the input, t ∈ R the time and l ( t ) ∈ R the time-varying lower bound for the state. Using the Pontryagin's maximum principle, I have defined the Hamiltonian as: H = ψ g ( x , u , t ) + f ( u ) and used the necessary condition ψ ˙ = − ∂ H ∂ x . As you can see I have completely ignored the time-dependent inequality, so the solution I get is correct, but of course it doesn't enforce the lower bound l on the state value.How can I rewrite the hamiltonian so that also the inequality is considered?

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need to build the optimal controller, i.e. one that maximizes:  J  =      ∫      0            t      f        f  (  u  )  d  tFor the following time-dependent system:      x    ˙    =  g  (  x  ,  u  ,  t  )where   x  (  t  )  ∈  R is the state,   u  (  t  )  ∈  R the input,   t  ∈  R the time and   l  (  t  )  ∈  R the time-varying lower bound for the state. Using the Pontryagin&#039;s maximum principle, I have defined the Hamiltonian as:   H  =  ψ  g  (  x  ,  u  ,  t  )  +  f  (  u  ) and used the necessary condition       ψ    ˙    =  −            ∂          H              ∂      x      . As you can see I have completely ignored the time-dependent inequality, so the solution I get is correct, but of course it doesn&#039;t enforce the lower bound   l on the state value.How can I rewrite the hamiltonian so that also the inequality is considered?

Paxton James · Accepted Answer

I think we need to add   λ  (  t  )  (  x  (  t  )  −  l  (  t  )  ) to the Hamiltonian, where   λ  (  t  ) is a time-dependent Lagrange multiplier. Then we need to find a triplet of functions   x  ,  u  ,  λ, such that the HJB-equations are satisfied and   λ  (  t  )  ≥  0 and   λ  (  t  )  =  0 if   x  (  t  )  &amp;gt;  l  (  t  ).

need to build the optimal controller, i.e. one that maximizes: J = <msubsup> &#x222B

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