Weierstrass conditions (for a variational extremum)

Necessary and (partially) sufficient conditions for a strong extremum in the classical calculus of variations (cf. Variational calculus). Proposed in 1879 by K. Weierstrass.

Weierstrass' necessary condition: For the functional

$$J(x)=\underset{t_0}{\overset{t_1}{\int }}L(t,x(t),\dot{x}(t))dt,L:\mathbf{R}\times {\mathbf{R}}^n\times {\mathbf{R}}^n\to \mathbf{R},$$

to attain a strong local minimum on the extremal $x_0(t)$, it is necessary that the inequality

$$\mathcal{E}(t,x_0(t),{\dot{x}}_0(t),\xi )\ge 0,$$

where $\mathcal{E}$ is the Weierstrass $\mathcal{E}$- function, be satisfied for all $t$, $t_0\le t\le t_1$, and all $\xi \in {\mathbf{C}}^n$. This condition may be expressed in terms of the function

$$\mathrm{\Pi }(t,x,p,u)=(p,u)-L(t,x,u)$$

(cf. Pontryagin maximum principle). The Weierstrass condition ( $\mathcal{E}\ge 0$ on the extremal $x_0(t)$) is equivalent to saying that the function $\mathrm{\Pi }(t,x_0(t),p_0(t),u)$, where $p_0(t)=L_{\dot{x}}(t,x_0(t),{\dot{x}}_0(t))$, attains a maximum in $u$ for $u={\dot{x}}_0(t)$. Thus, Weierstrass' necessary condition is a special case of the Pontryagin maximum principle.

Weierstrass' sufficient condition: For the functional

$$J(x)=\underset{t_0}{\overset{t_1}{\int }}L(t,x(t),\dot{x}(t))dt,L:\mathbf{R}\times {\mathbf{R}}^n\times {\mathbf{R}}^n\to \mathbf{R},$$

to attain a strong local minimum on the vector function $x_0(t)$, it is sufficient that there exists a vector-valued field slope function $U(t,x)$( geodesic slope) (cf. Hilbert invariant integral) in a neighbourhood $G$ of the curve $x_0(t)$, for which

$${\dot{x}}_0(t)=U(t,x_0(t))$$

and

$$\mathcal{E}(t,x,U(t,x),\xi )\ge 0$$

for all $(t,x)\in G$ and any vector $\xi \in {\mathbf{R}}^n$.

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See also Weierstrass–Erdmann corner conditions, for necessary conditions at a corner of an extremal.

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