View source for Optimality, sufficient conditions for

Conditions which ensure the optimality of a given solution of a problem of the variational calculus in a chosen class of comparison curves.

Sufficient conditions for optimality of a weak minimum (see [[#References|[1]]]): For a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o0685301.png" /> to provide a weak minimum for the functional

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o0685302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

given the boundary conditions

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it is sufficient that the following conditions are fulfilled.

1) The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o0685304.png" /> must be an extremal, i.e. it must satisfy the [[Euler equation|Euler equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o0685305.png" /></td> </tr></table>

2) Along the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o0685306.png" />, including its ends, the strong [[Legendre condition|Legendre condition]]

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must be fulfilled.

3) The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o0685308.png" /> must satisfy the strong [[Jacobi condition|Jacobi condition]], which requires that the solution of the Jacobi equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o0685309.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

with initial conditions

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must not vanish at the points of the right-closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853011.png" />.

The coefficients of the Jacobi equation (2), which is a linear differential equation of the second order, are calculated along the extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853012.png" /> and represent known functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853013.png" />.

For a strong minimum it is sufficient that the following extra condition be fulfilled, next to those mentioned above.

4) There exists a neighbourhood of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853014.png" /> such that at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853015.png" /> of it, and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853016.png" />, the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

hold, where

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is the Weierstrass function, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853020.png" /> is the slope of the field of extremals surrounding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853021.png" />.

At the extremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853022.png" />, condition (3) takes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

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Condition (4) is necessary for a strong minimum; it is called the Weierstrass necessary condition (cf. [[Weierstrass conditions (for a variational extremum)|Weierstrass conditions (for a variational extremum)]]). Thus, unlike the sufficient conditions for a weak minimum, which require certain strong necessary conditions to be fulfilled at the points of the extremal, the sufficient conditions of a strong minimum require that the Weierstrass necessary condition be fulfilled in a neighbourhood of the extremal. In general, it is impossible to weaken the formulation of the sufficient conditions for a strong minimum once the requirement that the Weierstrass condition be fulfilled in a neighbourhood of the extremal has been replaced by the strong Weierstrass condition (condition (4) with strict inequality) at the points of the extremal (see [[#References|[1]]]).

For non-classical variational problems, examined in the mathematical theory of optimal control (cf. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), several approaches to the establishment of sufficient conditions for optimality of an absolute extremum exist.

Let a problem of optimal control be posed in which the minimum of the functional

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

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has to be determined, given the conditions

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853030.png" /> is a given closed set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853031.png" />-dimensional space.

Using the method of [[Dynamic programming|dynamic programming]] [[#References|[3]]], sufficient conditions for optimality are formulated in the following way. For a control <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853032.png" /> to be an optimal control in the problem (5)–(8), it is sufficient that:

a) there exists a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853033.png" /> which has continuous partial derivatives for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853034.png" /> with the possible exception of a certain piecewise-smooth set of dimension less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853035.png" />, which vanishes at the end-point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853037.png" />, and which satisfies the [[Bellman equation|Bellman equation]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853039.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853041.png" /> is a synthesizing function (cf. also [[Optimal synthesis control|Optimal synthesis control]]), which can be defined using the Bellman equation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853042.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853043.png" /></td> </tr></table>

In reality, when using the method of dynamic programming, a stronger result is obtained: Sufficient conditions for optimality for a set of different controls which transfer a phase point from an arbitrary initial state to a given final state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853044.png" />.

In the more general case of a non-autonomous system, i.e. the integrand and the vector-function of right-hand sides also depend on the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853045.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853046.png" /> will depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853047.png" /> and a term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853048.png" /> must be added to the left-hand side of equation (9). There is a proof (see [[#References|[4]]]) which assumes the condition of continuous differentiability of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853049.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853050.png" />, which is very restrictive and is not fulfilled in most problems although it is usually supposed.

Sufficient conditions for optimality can be constructed on the basis of the [[Pontryagin maximum principle|Pontryagin maximum principle]]. If in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853051.png" /> of the phase space a regular synthesis is realized (cf. [[Optimal synthesis control|Optimal synthesis control]]), then all trajectories obtained using the maximum principle when constructing the regular synthesis are optimal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853052.png" />.

The definition of a regular synthesis, although rather cumbersome, does not essentially impose any particular restrictions on the problem (5)–(8).

There is another approach to the construction of sufficient conditions for optimality (see ). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853053.png" /> be a continuous function with continuous partial derivatives for all admissible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853054.png" /> belonging to a given region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853055.png" />, and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(10)</td></tr></table>

For the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853057.png" /> to establish an absolute minimum in the problem (5)–(8), it is sufficient that there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853058.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(11)</td></tr></table>

Corresponding variations of the above-mentioned formulation of the sufficient conditions for optimality are permissible for the more general cases of non-autonomous systems, problems with Mayer- and Bolza-type functions (see [[Bolza problem|Bolza problem]]), as well as for optimal sliding regimes (see [[Optimal sliding regime|Optimal sliding regime]] and ).

Research has been carried out into variational problems with functionals in the form of multiple integrals and with differential constraints in the form of partial differential equations, in which functions of several variables occur (see [[#References|[6]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,   L.A. Lyusternik,   "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.A. Bliss,   "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Bellman,   "Dynamic programming" , Princeton Univ. Press  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.G. Boltyanskii,   "Mathematical methods of optimal control" , Holt, Rinehart &amp; Winston  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top">  V.F. Krotov,   "Methods for solving variational problems on the basis of sufficient conditions for an absolute minimum I"  ''Automat. Remote Control'' , '''23'''  (1963)  pp. 1473–1484  ''Avtomat. i Telemekh.'' , '''23''' :  12  (1962)  pp. 1571–1583</TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top">  V.F. Krotov,   "Methods for solving variational problems on the basis of sufficient conditions for an absolute minimum II"  ''Automat. Remote Control'' , '''24'''  (1963)  pp. 539–553  ''Avtomat. i Telemekh.'' , '''24''' :  5  (1963)  pp. 581–598</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.G. Butkovskii,   "Theory of optimal control of systems with distributed parameters" , Moscow  (1965)  (In Russian)</TD></TR></table>



====Comments====
One distinguishes fixed end-time and free end-time problems. The problem expressed by the equations (5)–(8) is a free end-time one; the end-time is determined by the moment that the state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853060.png" /> equals the final state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068530/o06853061.png" />.

For additional references see also [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Markus,   "Foundations of optimal control theory" , Wiley  (1967)</TD></TR></table>