Time optimal control problems for some non-smooth systems

Time optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin's maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin's maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one get a chance to get Pontryagin's maximum principle for the original optimal classical control. Existence results are also considered.

In this paper, the non-smoothness means that f may singular at target set. More precisely, we will consider cases of what f is locally bounded while f y may unbounded near the target set.
(1.4) Remark 1.1. We can see that the description above is a little different from a general one, such as the state function belongs to C([0, T ); R n ) but not C([0, T ]; R n ). On the other hand, we use lim s→T − d(y(s), Q) = 0 instead of y(T ) ∈ Q to describe that the state "reaches" the target.
That is to avoid unnecessary inconvenience. Under assumptions of this paper, it is possible that This paper is stimulated by Lin and Wang [9] for optimal blowup time problems, and Lin [8] for optimal quenching time problems. In [8], f itself is unbounded near the target. While in [9], the "target set" is {∞}, and f is also unbounded near the target. We observed that both optimal quenching time problems and optimal blowup time problems can be transformed to time optimal control problems for some non-smooth systems (see Problem (T)) by some suitable state transform.
As far as we know, the concept of quenching appeared in H. Kawarada [7] at the earliest, which studied the initial boundary value of the diffusion equation          u t = u xx + 1 1 − u , t > 0, x ∈ (0, l), u(t, 0) = u(t, l) = 0, t > 0, u(0, x) = 0, x ∈ (0, l). (1.5) The author give the definition of quenching: namely, u can get to 1 in finite time. The author also get some sufficient conditions of the quenching behavior.
The quenching phenomenon can be observed in the transient current of polarized ion conductor . The quenching of solution means that the derivative of the solution goes to infinity in finite time while it keep bounded itself. Many people are interested in quenching behavior of partial differential equations. See [2], [3], [6] for the reference.
Blowup is another concept related to quenching which means the solution is unbounded in finite time. The difference between them is that solution keep bounded at quenching time while explode at the blowup time. We refer the readers the following works on blowup behaviors: [1], [4], [5], [6], [11] and [12], for examples.
Our aim is to study the existence theory and the necessary conditions for solutions of Problem (T). The non-smoothness bring much more difficulties than what we though at the beginning. In this paper, we get some existence results but only establish Pontryagin's maximum principle for one of relaxed optimal controls for general systems. Nevertheless, we think that for many special systems, by using such a result and the monotonicity of controlled systems, we can finally establish Pontryagin's maximum principle for any optimal control of Problem (T). Now, let us state the problem more precisely. We make our basic assumptions as follows: (S1) Set U ⊂ R m is a non-empty bounded closed set; (S2) Set Q is a non-empty convex closed set in R n , y 0 ∈ Q; (S3) Function f (t, y, u) is measurable in t ∈ [0, +∞) and continuous in (y, u) ∈ (R n \Q)×U .
Moreover, for any E ⊂⊂ ×(R n \ Q), there exists a constant L E > 0 and uniform modulus of As mentioned above, y(·) is not always well-defined after it reaches Q. Since we only concern about the first time when it reaches Q, and y(·) is uniquely determined by (1.1) before that time from (S3)-(S4), we can denote the solution of (1.1) by y(·; u(·)) without any misunderstanding.
The outline of this paper is: in first section, we give a general description of the problem. In the second section, we explain the problem in detail and make some transformation. The third section will devote the relaxed control. The fourth section will give the existence of the optimal control, and the fifth section will focus on the maximum principle of the optimal control.
2 Relaxed Problems and the Corresponding Results.
To study the existence of optimal control, we introduce the relaxed control which can also help to study the necessary conditions of optimal controls. To deal with the non-smoothness of the system near the target set, we need to introduce the corresponding approximate problem.
However, approximate problems need not necessary admits optimal control even if the optimal control of the original problem exists. To overcome this difficulty, we consider relaxed controls.

Relaxed controls.
Now, we recall the notion of relaxed control and state some preliminary results about the space of relaxed controls.
We denote by M 1 + (U ) the set of all probability measures in U , by R(U ) (R T (U ) ) the set of all measurable probability measure-valued functions on [0, +∞) ([0, T ]), that is, σ(·) ∈ R(U ) (R T (U ) ) if and only if where C(U ) denotes the space of all continuous functions on U . Let C(U ) * and L 1 ([0, T ]; C(U )) * be the dual spaces of C(U ) and L 1 ([0, T ]; C(U )) with weak star topology, respectively. We regard M 1 + (U ) and R T (U ) as subspaces of C(U ) * and L 1 ([0, T ]; C(U )) * , respectively, by setting We see that (2.2) is well-defined by Theorem IV.1.6, (p. 266) in [10]. We say that while say that In fact, we only concern R T (U ) for T large enough, but not R(U ) itself. We introduce R(U ) just for the convenience as the optimal time is unknown for us.
The following lemma is an important property of relaxed controls.

Existence of optimal relaxed controls.
First, we study the existence of the optimal relaxed controls. Consider the following relaxed controlled system corresponding to (1.1): Sets RP, RP ad and R ad are called the set of feasible relaxed triples , the set of admissible relaxed triples and the set of admissible relaxed controls.
We give the following lemma, which concerns the continuous dependence of the solutions of (2.3) with respect to the relaxed controls in the meaning of convergence in R(U ).
Proof. By Lemma 2.1, we can suppose that (2.14) Then, it follows easily from Lemma 2.2 that there must exist S ∈ (0, w] such that y(·; σ(·)) exists on [0, S) and That is, (S,ŷ(·),σ(·)) ∈ RP ad . Therefore the optimal time of Problem (R) is not bigger than The lemma above says that if the optimal time of Problem (R) is t * , then for any S ∈ (0, t * ), and σ(·) ∈ R, the solution of (2.3) exists on [0, S]. Moreover, the distance between and Q is positive, that is, inf The following theorem gives the existence of optimal relaxed triples under relatively weak assumptions.
Then Problem (R) admits at least one optimal relaxed triple.

Maximum Principle for Optimal Relaxed Triple.
To yield the maximum principle for optimal relaxed triple, we introduce approximate problems.
For any α > 0, denote Then Q α is closed and convex. Moreover, denote We introduce the following approximate problems.
We have the following result.

Moreover, there exists a nontrivial solution of
In the above lemma, the existence results can be looked a corollary of Theorem 2.4, while the necessary conditions can be yielded in a standard way like that for optimal classical triple in smoothness cases.

Remark 2.3.
Let Ω ⊃⊃ Q α and G ∈ C 1 (Ω; R n ). Moreover, ∂G(x) ∂x is not singular for any x ∈ Ω. If RP α ad is replaced by then Lemma 2.5 holds with (2.16) being replaced by The following theorem is related to the necessary conditions for optimal relaxed controls.
To get the transversality condition, we set the following assumptions.
Combining the above, we can get that for any s 1 , s 2 ∈ [1 − δ, 1], it always holds that is deducted from (S3 ′ ), but have nothing to do with (CE1)-(CE2). Therefore, (S3 ′ ) can be replaced by the following: (S3 ′′ ) There exists a homeomorphic mapping x = x(y) such that is linear increasing in x .
Remark 2.5. The singularity of f near Q bring many difficulties in yielding maximum principle.
Though we think the maximum priciple should be hold for any optimal relaxed triple, we failed to got such a result in a general way. Nevertheless, we think it is possible to use the above result to yield maximum priciple for an optimal classical control for many special controled systems.

Existence of Optimal Classical Control.
To get the existence of optimal classical triple, we set the following assumption.
(ES) Assume that for any (t, y) ∈ [0, +∞) × R n \ Q, {f (t, y, u)|u ∈ U } is closed and convex. If RP ad = ∅, in particular, if P ad = ∅, then Problem (T) admits at least one optimal classical triple. Moreover, any optimal classical triple is also an optimal relaxed triple.

Maximum Principle for Optimal Classical Control.
It is only in the case that an optimal control to Problem (T) is the unique optimal relaxed control to Problem (R), we can get the corresponding maximum principle directly from Theorem 2.6. For many particular systems, given an optimal classical triples (t,ȳ(·),ū(·)), it is possible for us to construct a suitablef , such that (t,ȳ(·),ū(·)) is the unique optimal relaxed triple to a new optimal control problem with Then, we can get that (t,ȳ(·),ū(·)) satisfies the maximum principle immediately from Theorem 2.6.
Nevertheless, unfortunately, we fail to prove the maximum principle for every optimal control in general situation.
If the system is affine, namely f (t, y, u) is in the form of g(t, y) + B(t)u, then we have the following result.

Optimal time control problems for some particular systems
In this section, we will show how to apply Theorem 2.6 to yield maximum principle for (every) optimal classical control for particular systems. We will discuss two examples considered in [8] and [9]. A crucial property we used here is the " monotonicity" of the controlled systems.
The first example is concerned optimal quenching time which is considered in [8].
where y 0 = y 10 We give first a lemma concerning the monotonicity of a system related to (6.2).
admitting positive measure. Denoteŷ 1 (·) as the solution of equation y 20 . Proof. We only prove (i) while (ii) can be proven similarly. The proof will be finished in two steps.
and the following optimal control problem: to find v(·) ∈ V such that z 1 (T ) takes the minimum value.
If y 2 (T ) > 0, then there exits ε ∈ (0, T − S) such that Therefore We get the proof. ✷ For the time optimal control corresponding to Example 1, we have the following theorem, which is an extension of Theorem 1.4 in [8].
We will get the results by introducing a new system such that the optimal classical triple in consideration becomes the unique optimal relaxed triple of the new optimal problem.
I. Consider that We call Problem (R) corresponding tof as Problem ( R).
On the other hand, one can easily see that for some C > 0. Consequently, Then t 0ȳ 2 (t) (1 −ȳ 1 (t)) 2 dt = +∞ and it follows from and L'Hospital's rule that That is, we get (6.20). ✷ Now, we consider an example concerning optimal blowup time which is considered in [9].
where M and r 0 are defined by (6.28) and (6.31), Proof. We will prove the lemma in two steps.
We can construct a smooth inversable map G : R n \ B β (X 0 ) → B R (0) such that where R > R 1 > Gb, γ > p − 1, and B r (X) denotes the ball of radius r and centered at X.