Singular control

In <a href="/facts/Optimal_control/9SmcQHjJ">optimal control</a>, problems of singular control are problems that are difficult to solve because a straightforward application of <a href="/facts/Pontryagin%2527s_minimum_principle/gEVJ1sjc">Pontryagin's minimum principle</a> fails to yield a complete solution. Only a few such problems have been solved, such as <a href="/facts/Merton%2527s_portfolio_problem/WtewGQk1">Merton's portfolio problem</a> in <a href="/facts/Financial_economics/oraavYoW">financial economics</a> or <a href="/facts/Trajectory_optimization/cjQAMobd">trajectory optimization</a> in aeronautics. A more technical explanation follows.
The most common difficulty in applying Pontryagin's principle arises when the <a href="/facts/Hamiltonian_(control_theory)/cPcyjLhU">Hamiltonian</a> depends linearly on the control 
 
 
 
 u
 
 
 {\displaystyle u}
 
, i.e., is of the form: 
 
 
 
 H
 (
 u
 )
 =
 ϕ
 (
 x
 ,
 λ
 ,
 t
 )
 u
 +
 ⋯
 
 
 {\displaystyle H(u)=\phi (x,\lambda ,t)u+\cdots }
 
 and the control is restricted to being between an upper and a lower bound: 
 
 
 
 a
 ≤
 u
 (
 t
 )
 ≤
 b
 
 
 {\displaystyle a\leq u(t)\leq b}
 
. To minimize 
 
 
 
 H
 (
 u
 )
 
 
 {\displaystyle H(u)}
 
, we need to make 
 
 
 
 u
 
 
 {\displaystyle u}
 
 as big or as small as possible, depending on the sign of 
 
 
 
 ϕ
 (
 x
 ,
 λ
 ,
 t
 )
 
 
 {\displaystyle \phi (x,\lambda ,t)}
 
, specifically:

u
 (
 t
 )
 =
 
 
 {
 
 
 
 b
 ,
 
 
 ϕ
 (
 x
 ,
 λ
 ,
 t
 )
 <
 0
 
 
 
 
 ?
 ,
 
 
 ϕ
 (
 x
 ,
 λ
 ,
 t
 )
 =
 0
 
 
 
 
 a
 ,
 
 
 ϕ
 (
 x
 ,
 λ
 ,
 t
 )
 >
 0.
 
 
 
 
 
 
 
 
 {\displaystyle u(t)={\begin{cases}b,&\phi (x,\lambda ,t)<0\\?,&\phi (x,\lambda ,t)=0\\a,&\phi (x,\lambda ,t)>0.\end{cases}}}

If 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a <a href="/facts/Bang-bang_control/YpKToXYI">bang-bang control</a> that switches from 
 
 
 
 b
 
 
 {\displaystyle b}
 
 to 
 
 
 
 a
 
 
 {\displaystyle a}
 
 at times when 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 switches from negative to positive. 
The case when 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 remains at zero for a finite length of time 
 
 
 
 
 t
 
 1
 
 
 ≤
 t
 ≤
 
 t
 
 2
 
 
 
 
 {\displaystyle t_{1}\leq t\leq t_{2}}
 
 is called the singular control case. Between 
 
 
 
 
 t
 
 1
 
 
 
 
 {\displaystyle t_{1}}
 
 and 
 
 
 
 
 t
 
 2
 
 
 
 
 {\displaystyle t_{2}}
 
 the maximization of the Hamiltonian with respect to 
 
 
 
 u
 
 
 {\displaystyle u}
 
 gives us no useful information and the solution in that time interval is going to have to be found from other considerations. One approach is to repeatedly differentiate 
 
 
 
 ∂
 H
 
 /
 
 ∂
 u
 
 
 {\displaystyle \partial H/\partial u}
 
 with respect to time until the control u again explicitly appears, though this is not guaranteed to happen eventually. One can then set that expression to zero and solve for u. This amounts to saying that between 
 
 
 
 
 t
 
 1
 
 
 
 
 {\displaystyle t_{1}}
 
 and 
 
 
 
 
 t
 
 2
 
 
 
 
 {\displaystyle t_{2}}
 
 the control 
 
 
 
 u
 
 
 {\displaystyle u}
 
 is determined by the requirement that the singularity condition continues to hold. The resulting so-called singular arc, if it is optimal, will satisfy the Kelley condition:

(
        −
        1
        
          )
          
            k
          
        
        
          
            ∂
            
              ∂
              u
            
          
        
        
          [
          
            
              
                
                  (
                  
                    
                      d
                      
                        d
                        t
                      
                    
                  
                  )
                
              
              
                2
                k
              
            
            
              H
              
                u
              
            
          
          ]
        
        ≥
        0
        ,
        
        k
        =
        0
        ,
        1
        ,
        ⋯
      
    
    {\displaystyle (-1)^{k}{\frac {\partial }{\partial u}}\left[{\left({\frac {d}{dt}}\right)}^{2k}H_{u}\right]\geq 0,\,k=0,1,\cdots }

Others refer to this condition as the generalized <a href="/facts/Legendre%25E2%2580%2593Clebsch_condition/P5EzN1eP">Legendre–Clebsch condition</a>.
The term bang-singular control refers to a control that has a bang-bang portion as well as a singular portion.

Singular control open-in-new

Singular control