24 Summary of Optimal Control Conditions
24.1 Canonical Bolza problem
Consider
\begin{aligned} \operatorname*{minimize}_{\bm{x},\bm{u},\bm{p}}\quad &J=\Phi\big(\bm{x}(a),\bm{x}(b),\bm{p}\big) +\int_a^bL(\bm{x},\bm{u},\bm{p},t)\,\mathrm dt,\\ \text{subject to}\quad &\bm{x}'=\bm{f}(\bm{x},\bm{u},\bm{p},t),\\ &\bm{b}\big(\bm{x}(a),\bm{x}(b),\bm{p}\big)=\bm{0},\\ &\bm{\Psi}\big(\bm{x}(a),\bm{x}(b),\bm{p}\big) +\int_a^b\bm{g}(\bm{x},\bm{u},\bm{p},t)\,\mathrm dt=\bm{0},\\ &\bm{u}(t)\in\mathcal U. \end{aligned}
Here \bm{p} denotes constant optimization parameters, if present.
24.2 Integral-constraint augmentation
Introduce \bm{z} by
\bm{z}'=\bm{g}(\bm{x},\bm{u},\bm{p},t), \qquad \bm{z}(a)=\bm{0}.
The isoperimetric constraint becomes the endpoint condition
\bm{\Psi}\big(\bm{x}(a),\bm{x}(b),\bm{p}\big)+\bm{z}(b)=\bm{0}.
Thus the integral constraint is not removed merely by introducing \bm{z}'; its terminal boundary condition is essential.
24.3 Hamiltonian and endpoint Lagrangian
Using the normal minimization convention, define
\begin{aligned} H(\bm{x},\bm{\lambda},\bm{\eta},\bm{u},\bm{p},t) &=L(\bm{x},\bm{u},\bm{p},t) +\bm{\lambda}^{\mathsf T}\bm{f}(\bm{x},\bm{u},\bm{p},t) +\bm{\eta}^{\mathsf T}\bm{g}(\bm{x},\bm{u},\bm{p},t),\\ B(\bm{x}_a,\bm{x}_b,\bm{p},\bm{\omega},\bm{\eta}) &=\Phi(\bm{x}_a,\bm{x}_b,\bm{p}) +\bm{\omega}^{\mathsf T}\bm{b}(\bm{x}_a,\bm{x}_b,\bm{p}) +\bm{\eta}^{\mathsf T}\bm{\Psi}(\bm{x}_a,\bm{x}_b,\bm{p}). \end{aligned}
The augmented functional may be written as
\mathcal J_a =B+\int_a^b \left[ H-\bm{\lambda}^{\mathsf T}\bm{x}' -\bm{\mu}^{\mathsf T}\bm{z}' \right]\mathrm dt +\bm{\eta}^{\mathsf T}\bm{z}(b).
Variation with respect to \bm{z} gives \bm{\mu}'=\bm{0} and \bm{\mu}(t)=\bm{\eta}, so the auxiliary costate is constant and can be eliminated.
24.4 Necessary conditions
A normal extremal satisfies
\boxed{ \begin{aligned} \bm{x}' &= \frac{\partial H}{\partial\bm{\lambda}} =\bm{f}(\bm{x},\bm{u},\bm{p},t),\\ \bm{\lambda}' &= -\frac{\partial H}{\partial\bm{x}},\\ \bm{b}\big(\bm{x}(a),\bm{x}(b),\bm{p}\big)&=\bm{0},\\ \bm{\Psi}\big(\bm{x}(a),\bm{x}(b),\bm{p}\big) +\int_a^b\bm{g}(\bm{x},\bm{u},\bm{p},t)\,\mathrm dt&=\bm{0},\\ \frac{\partial B}{\partial\bm{x}_a}+\bm{\lambda}(a)&=\bm{0},\\ \frac{\partial B}{\partial\bm{x}_b}-\bm{\lambda}(b)&=\bm{0},\\ \frac{\partial B}{\partial\bm{p}} +\int_a^b\frac{\partial H}{\partial\bm{p}}\,\mathrm dt&=\bm{0},\\ \bm{u}^\star(t)&\in \operatorname*{arg\,min}_{\bm{v}\in\mathcal U} H\big(\bm{x}^\star(t),\bm{\lambda}(t),\bm{\eta},\bm{v},\bm{p},t\big). \end{aligned}}
If the optimal control is interior to \mathcal U and H is differentiable in \bm{u}, the last condition implies
\frac{\partial H}{\partial\bm{u}}=\bm{0}.
It should not be imposed as an additional equation for a boundary control. For convex \mathcal U, the equivalent first-order variational inequality is
H_{\bm{u}}(\bm{u}^\star)^{\mathsf T}(\bm{v}-\bm{u}^\star)\ge0, \qquad \forall\bm{v}\in\mathcal U.
24.5 Free terminal time
If the terminal time T is free, define
B(\bm{x}_0,\bm{x}_T,T,\bm{\omega}) =\Phi(\bm{x}_0,\bm{x}_T,T) +\bm{\omega}^{\mathsf T}\bm{b}(\bm{x}_0,\bm{x}_T,T).
The additional transversality condition is
H(T)+\frac{\partial B}{\partial T}=0.
For an autonomous problem with no explicit T-dependence in B, this reduces to H(T)=0.
24.6 Singular controls
For a scalar control appearing linearly,
H=A(\bm{x},\bm{\lambda},t)+\sigma(\bm{x},\bm{\lambda},t)u,
where \sigma=H_u is the switching function. With u\in[u_{\min},u_{\max}],
u^\star=\begin{cases} u_{\min},&\sigma>0,\\ u_{\max},&\sigma<0. \end{cases}
If \sigma\equiv0 on an interval, the arc is singular. Differentiate \sigma along the extremal until the control appears explicitly; then verify the appropriate higher-order necessary condition, such as the generalized Legendre–Clebsch condition, before claiming local optimality.
24.7 Numerical solution viewpoint
Indirect methods solve the state–costate boundary-value problem generated by the necessary conditions. Direct methods instead parameterize or collocate the trajectories and solve the resulting sparse nonlinear program. In either case, a numerical result should be checked for dynamic consistency, endpoint and path feasibility, optimality residuals, and stability under mesh refinement. The formulations and diagnostic procedures are developed in Numerical Methods for Optimal Control.
The equations above are necessary conditions. Existence, uniqueness, and sufficiency require additional hypotheses. Abnormal extremals require the more general Hamiltonian H=\lambda_0L+\bm{\lambda}^{\mathsf T}\bm{f} with \lambda_0\ge0 and (\lambda_0,\bm{\lambda}) nontrivial.