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Âûï. 4. - 2015. - URL:
Equivalent substitution in the control theory
© 2015 ã. Proudnikov I. M.
In this paper a system of the differential equations with a control is considered. We study a problem of looking for an optimal control that gives an infimum for an optimized functional. The system of differential equations is replaced by two systems with the upper and lower envelopes of a function on the right hand side of the initial system of the differential equations. The optimized functional is replaced by its lower envelope. All replacements are done in a region of attainability. The necessary conditions of optimality are sufficient for the substituted system. The rules for evaluation of the attainability set with the help of positively definite functions are given in the second part of the paper.
Key words. Optimal control, optimal trajectories, convex functions, lower and upper convex envelopes, attainability set, convex analysis, linear and convex functions.
Consider the following general problem of the control theory. Suppose we have a system of the differential equations
, and an optimized functional has a form
where , takes values in , where is a convex compact set in , . We assume that the function is continuous and the function is a Lipschitz one in all arguments in totality, so that the system (1) satisfies the conditions of uniqueness for a solution with the given initial values. We consider the autonomous systems of the differential equations it does not restrict the generality of consideration, as soon as if in the non-autonomous case is Lipschitz in all arguments uniformly in , then all argumentations, made below, will be true as well.
We have to find an optimal control that is a piecewise continuously differentiable vector-function from with values in . We will assume that the derivatives , where they exist, are bounded in the norm, i.e.
uniformly in . Here is a set of points in , where the derivatives exist. In this case pointwise convergence of a sequence on is equivalent to uniform convergence of the functions on a set of continuity and, of course, is equivalent to convergence in the metrics of the space . The metrics of is equal, by definition, to the metrics of the space , i.e.
The function is defined on the segment . We are looking for for that a solution to the system (1) gives an infimum for the functional .
In the beginning we replace the optimization problem (2) by the following problem
Let us include into consideration all the functions resulting from pointwise convergence. It is obvious that all limit functions belong to a closed, bounded set of functions defined on , which we denote by . The two functions from the set are equivalent (equal) if these functions are equal on a set of full measure. It is clear that all measurable functions on belong to .
We will solve the above formulated optimization problem (3) on the set , i.e.
The problem is that an optimal control does not exist always. For this reason generalized control (lower or upper semicontinuous) is considered.
As an example, consider the following system of differential equations
and the optimized functional is defined as
The similar problem we can in  with replacement on . This problem does not have an optimal control in the set of piecewise continuously differentiable functions on , but it has an optimizing sequence of controls that are piecewise continuous functions with values . It is easy to see that an optimizing sequence , corresponding to the sequence of controls , has the limit on . The control , that corresponds to the solution , can not be received as the pointwise limit of and it’s not an optimal control.
The right-hand side of the equation (1) can be very complex, and the exact solution of this equation can often be found approximately using numerical methods. Optimization of the function is also not easy if it has a complex form. But optimization of the lower convex envelope (LCE) of is easier. Moreover, a global optimum point does not disappear if we construct a lower convex envelope of our optimized functional. In addition, the construction of the lower convex envelope of , that we denote by , turns it into a lower weakly semicontinuous function. It means that
for any sequence converging to weakly. This requirement is important for weak convergence of an optimizing sequence to a solution of the problem (2).
We propose here a method of equivalent substitution with the help of which we can overcome these difficulties. Namely, we suggest a replacement of the right-hand side of the equation (1) by another function with a simpler structure. The search for solutions of the system (1) (numerical or not) becomes simpler. The principle of equivalent replacement claims that although we have another function with a simpler structure, but the function attains the same infimum on the set of piecewise continuously differentiable functions and on its closure. At the same time the new optimized functional becomes lower semicontinuous.
This idea is different from the idea of relaxation  because we take the convex (concave) envelopes of functions on the right hand side of our system of the differential equations for all points from the region of attainability.
Taking into consideration the information about the replacements of the functions and , we can conclude that searching for an optimal control and optimal trajectory becomes easier and the new optimization problem is equivalent to the initial one in the sense of finding an optimal control. In this case the conditions of optimality become necessary and sufficient.
For the first time the author used the idea about replacing a function with its lower convex approximation for finding its optimal points in  -.
Let us consider the same system of differential equations (1), and the optimized functional (4). We rewrite the system (1) and (4) in the following form
with the initial conditions . The optimization problem (4) is replaced by another optimization problem
Any solution of (5) is a solution of the integral equations
where is a piecewise continuously differentiable control.
Unite all solutions of (7) in one set for
which is called the set of attainability for the systems (1) and (4) at time .
It is easy to see that the optimization problem (6) is equivalent to the following optimization problem
The function is linear in the coordinates . (The function depends only on the coordinate ). It is well known that any linear function reaches its maximum or minimum on boundary of any compact set on which maximum or minimum are looked for.
Since the set of solutions of (1) in accordance with the assumptions is bounded on , the set of the vector-valued functions is closed and bounded in . Then is closed and bounded for any in the metrics of the space . Indeed, if , then, as it was mentioned above, there is uniform convergence of to on , where is a set of any small measure. Then convergence in measure  holds. The convergences
follow from continuity of the functions , in all variables.
Uniform convergence of the integrals
in and follows from Egorov’s theorem .
Indeed, otherwise the sequences and exist, for that and for some , the inequalities
hold. The integrals can be considered as functions of .
According to Egorov’s theorem for any small there is a set with measure , that the integrals, as the functions of , will converge uniformly in on a set . As soon as the integrals are absolutely continuous in measure, the integrals over the set with measure will be arbitrarily small if is also arbitrarily small. As a result, we come to the contradiction with existence of , for which the inequalities, written above, are true. Thus we have proved the following theorem.
Theorem 2.1 The set is closed and bounded in for any in the metrics of the space .
Consider a sequence of the functions defined on ,
The sequence converges on uniformly in , if the sequence converges in the metrics to the function a.e. on . Prove this fact.
Indeed, we know from the said above that the functions converge to uniformly on . We replace the control by the control in (10). The difference between the original value of the integral (10) and the new value of the same integral can be evaluated in the following way. According to the inequality
where is a Lipschitz constant of the function , the mentioned above difference is arbitrarily small for large as well. Indeed, we have
and the right hand side of this inequality is arbitrary small for large .
We will use the following result.
Lemma 2.1 , . The sequence converges uniformly for and to a solution of (1).
Summing up everything mentioned above, we can conclude about uniform convergence on of the solutions of (1) for to a solution of the same system (1) with the control as .
Lemma 2.2 The sequence defined by (10), converges uniformly on in to a solution of (1).
Remark 2.1 Lemma (2.2) is also valid for the case when in the metrics of the space , i.e.
The problem (9) has a solution if the functional (2) is lower semicontinuous. It will be shown how to make it lower semicontinuous.
If there is a solution to the problem (9) on the set of piecewise continuously differentiable functions , then we have a solution to the problem
where is a symbol of taking convex hull.
We introduce a set of attainability (or an attainability set) for the time , which, by definition, is
where means closed convex hull. It is easy to see that for an arbitrary
will be true. Therefore, the closure in (12) can be removed and definition of the set can be given as the following
Moreover, the problems (9) and (11) are equivalent that means: if one of them has a solution, then the other one has a solution as well and these solutions are equal to each other. In addition, since projections of the set on the axes are closed and bounded, and, hence, compact in the corresponding finite-dimensional spaces and is continuous in as the set-valued mapping, then in (11) can be replaced by and the problem (11) can be rewritten in the following way
But a global optimal point of the problem (14) will not change if we replace the functions by their upper concave and lower convex envelopes and by its lower convex envelope constructed on a set of attainability for the time . Further we will understand under taking the upper concave and lower convex envelopes of the similar operations for all coordinates of .
Indeed, take two arbitrary points and from the set . Consider a combination with nonnegative coefficients , , . Then, the point will belong to the set , if we replace the functions and by the following:
But this construction, performed for all points of the regions it just means that we construct the lower and upper convex envelopes of the function and the lower convex envelope of the functions in the attainability set for the time , i.e. in . Indeed, it follows from the above formula (13) for that the function reaches its minimum (14) on some . Consequently, we can construct the lower convex or upper concave envelopes of the functions and in all region .
Denoted by a new optimization function obtained after the replacement of the function by in
It is clear that takes the same optimal value in the attainability set , that the functional takes for the system (1).
Replace the system by the system
and the optimized functional by the functional
It is easy to see that the minimum or the maximum of the functional did not change. Hence, the problems , and , are replaceable. So "convexification" of the function , in contrast to the procedure of "convexification" of the function should be as following:
1. Construction of the lower convex envelope (LCE) of the function in the variables for each from the attainability set for the time , i.e. , which we denote by . LCE of is the biggest convex function that does not exceed in .
2. Construction of the upper concave envelope (UCE) of the function (or, equivalently, we construct the lower convex envelope for the function and after that take minus of this function) in the variables for each from the attainability set for the time , i.e. , which we denote by . UCE of is the smallest concave function that is not less in .
3. Let us replace the system , by two systems of the equations:
with the optimization function and
with the same optimization function ;
4. Let us find among the solutions of and such that gives the smallest value of the functional in .
We obtain the following result.
Theorem 2.2 There are the solutions among the solutions of and such that deliver a minimum (maximum) in and for the functional
that coincides with an infimum (supremum) of the functional (see (4)). Moreover, necessary conditions for the minimum (maximum) are also sufficient conditions.
Remark 2.2 The set is not necessarily compact, although its projections on the axis are compact . That’s why we are able to go to the problem
if the problem (9) has a solution. The last one coincides with the formulation of Mazur’s theorem. It asserts that in any weakly convergent sequence , a subsequence can be chosen for each convex hull of which is almost everywhere on converges as to some . In our case, there exists a sequence , the convex hull of which will converge to an optimal control . The sequence of the solutions , corresponding to the controls , will converge to an optimal solution , corresponding to the control , provided that the solutions have been calculated to the problems with the modified right-hand side.
Remark 2.3 The rules for construction of LCE and UCE are given in Appendix.
Remark 2.4 In many cases we have to construct only LCE or UCE for the function .
Return back to the initial problem (2) with the fixed time . Consider a set
that is called the set of attainability of the system (1),(2) at time .
Let us introduce a set of attainability for the time for the system (1),(2) that is by definition
As above it is possible to prove that we can remove the closure in (19) and write
The optimization problem can be reformulated in the form
The problems (1), (2) and (20) are equivalent which means if one has a solution, then another one has a solution and these solutions are the same. Moreover, as soon as the projections of the sets on the axes are closed, bounded and continuous as a set valued mappings, then we can write instead of if a solution of (20) exists.
We come to the following result.
Theorem 2.3 There are some solutions among the solutions of and such that deliver a minimum (maximum) in and to the functional
that coincides with an infimum (supremum) of the functional (see (2)) where .is LCE of the function Moreover, the necessary conditions for minimum (maximum) are also the sufficient conditions.
Consider some examples. It is clear that an equivalent replacement of one system by another can be applied to a differential system without control .
Example 1. Consider the differential equation
with the initial condition . The optimized functional is given by
The general solution of the differential equations for has the form
which tends to as . The general solution of the differential equation for is given by
which tends to as . In order to meet the initial condition we have to put . The projection of the attainability set on axis is the interval
It is clear that the function takes its minimum at . But we get the same solution if instead of the function we take its lower convex envelope, namely, the function
Example 2. The same example, but
which we minimize for . The equation of the solution is
Here the replacement of the function by the function on the whole line is not correct, since the projection of the attainability set on axis is the interval .
Example 3. Let us give the differential equation
with the initial condition . The general solution has the form
a solution, satisfying the initial condition, is
The optimized functional has the form
It is easy to compute its optimal value
In this case, the projection of the attainability set on axis is the set . It is easy to see that the lower convex envelope of the functional on , which we denote by , takes the same infimum value. It is also true for the functional
Example 4. Let us consider the following problem
The optimized functional is
We will get the following system after construction of the lower and upper envelopes
The optimal solution exists among their solutions .
Example 5. Let us consider the differential equation
with the initial condition . We are considering piecewise continuously differentiable functions on the segment for such that delivers minimum to the functional
The solution when is constant on the segment is given by the form
Here the constant is defined by the initial conditions. We can see from here that if is not constant on , then for any initial conditions. It means that a curve will be in a set bounded by the lines on the plane , where . The set of attainability will belong to the same set. UCE of the function in is a function the graph of which goes through the point . Therefore, if we solve the differential equations with the right sides and , then among the solutions there are such that deliver the minimum to the functional i.e. the formulated theorem is true.
Let us have a system of differential equations
with the initial condition , where is Lipschitz in the variables , is a convex compact set in . The problem is to estimate the attainability set. By definition, the area of attainability for the time is the set
The choice of the initial position and the initial time of zero is not a loss of generality.
Take an arbitrary positively definite function (see ), satisfying the condition
and is a piecewise continuous vector-function.
Consider the systems of differential equations
We denote by
the attainability sets for the systems (22), (23), where
Let the estimates of the attainability sets for the time be given respectively by the inequalities
We get the estimation for the attainability set of the system (21) for the time . We show that the attainability set for this system satisfies the inclusion
Indeed, by definition, the set will consist of the points on the curves the tangents to which are the sum of the tangents to the curves consisting of the points of the sets and for all . It is clear, that for some vector-function the resulting set will include , which consists of the points on the curves the tangents to which are the sum of the tangents to the curves consisting of the points of the sets and for all .
As a result, the following theorem is proved.
Theorem 3.1 For the attainability set of (21) the inclusion
is true for some vector-function , where , are given by (24), (25) .
The following lemma follows from here.
Lemma 3.1 The function satisfies the inequality
in the attainability set of the system (21).
Now consider two differential systems with the right sides , and zero initial conditions at the time equaled to zero.
Let us suppose that the inequality
is true for all . We assume that we know the attainability set for the time of the system with the right hand side . The problem is to obtain some estimates of the attainability set of the system (21). The arguments will be carried out as previously, considering the trajectories of the corresponding systems.
Any vector in the set for some and is
Since the previous inclusion holds for any , it follows that
The following theorem is proved.
Theorem 3.2 For the systems (21) and (23) with the attainability sets and respectively, for which the inequality (28) holds, the inclusion
From here we can easily obtain the following conclusion.
Lemma 3.2 In the attainability set of the system (21) the function satisfies the inequality
where the constant limits the top value of the function in the attainability set of the system (23).
Let us give a general method for evaluation of of the system (21). This method does not require any additional information for the system (21).
As is known, a convex set can be given by its extreme points. There are no problems if there is a finite number of such points. But very often these points are unknown or their number is infinite. We can reconstruct a convex set if we know its projections on different directions. If we project any trajectory, then we project not only the points but the tangents constructed at these points. It means that we have to consider the following system for any direction
As a result, we have
where is the scalar production .
We can do an orthogonal transformation that the direction of the vector was the first coordinate axis . Then the projection of the velocity vector on the line will be equal to . It means that we have to substitute into the first equation and to solve the first order differential equation for different values of the constants that are corresponding coordinates of the start point.
We can make the following conclusion: convex hull of the attainability sets of the equation (27) for different will include the attainability set of the equation (21). It is possible to do, because calculation methods are developed very well for the first order differential equations.
The obtained results allow us to pass from local to global optimization problem. To implement this it is required to construct the lower convex and upper concave envelops of the function written on the right hand side of the differential system (1). We also construct the lower convex envelops for the optimized functional. All constructions are done in the attainability set for the time .
A method for estimation of the attainability set with the help of the positively definite functions (Lyapunov functions) is suggested. The proposed method is based on the decomposition of the function, stayed on the right hand side of the system of the differential equations, into the components the sets of attainability of which are already known. It makes it different from the paper , where the linear systems are considered.
It is suggested to find projections of onto any direction . For this reason we have to find projections of the trajectories of the differential system and the tangents to them to the direction . We come across the problem of definition of a set of attainability for the differential equation of the first order.
The proposed transformation method of the systems is especially useful when it is difficult to get a solution of differential equations in an explicit form, but while using approximate methods only. In addition, the sufficient conditions of optimality for an optimal control are obtained according to the proposed method.
We will prove a theorem giving a rule for construction of LCE and UCE.
Let be continuous function on a convex compact set . It is required to construct LCE and UCE in . Consider a function
where is a distribution function satisfying the following equalities
We will consider the functions for different distributions .
Theorem 4.1 The functions
are UCE and LCE of on correspondingly.
Proof. Without loss of generality we will consider that for all . Divide into subsets , . We can approximate the function with any precision by the integral sums
It follows from (28) that
The sign means that the values on the left hand side from this sign can be close to the values on the right hand side with any precision depending on . The expression (29) means that we take a convex hull of vectors with coefficients , i.e. we calculate a vector
and define a value of the function at this point equaled to
Changing the points and the coefficients satisfying (30), we define in such way the functions with different values at .
Let us prove that the function
is LCE. As soon as is taken for all distributions , then the inequality is true for all . The function can be approached by the sums (29) for any distribution under conditions on the coefficients (30). It follows from here that can not be smaller than LCE of . The operation keeps this quality. Consequently, is LCE of . We can prove in the same way that is UCE of . The Theorem is proved. ∆
The construction of LCE can be done using Fenchel-Morrey’s theorem . According to it LCE is equal to the second conjugate function . Construction of is not easy. To find a value of LCE at one point we have to solve two difficult optimization problems, namely,
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