# Abstract McKean-Vlasov and HJB equations, their fractional versions and
related forward-backward systems on Riemannian manifolds ††thanks: The work of
V.N. Kolokoltsov (Sections 1-7) was supported by the Russian Science
Foundation (project No. 20-11-20119), the work of M.S. Troeva (Sections 8-10)
was supported by the Ministry of Science and Higher Education of the Russian
Federation (Grant No. FSRG-2020-0006).

Vassili N. Kolokoltsov and Marianna Troeva Department of Statistics,
University of Warwick, Coventry CV4 7AL UK, and National Research University
Higher School of Economics, Pokrovskii bul. 11, Moscow, Russia., Email:
<EMAIL_ADDRESS>Institute of Mathematics, North-Eastern
Federal University, 58 Belinskogo str., Yakutsk 677000 Russia, Email:
<EMAIL_ADDRESS>
###### Abstract

We introduce a class of abstract nonlinear fractional pseudo-differential
equations in Banach spaces that includes both the Mc-Kean-Vlasov-type
equations describing nonlinear Markov processes and the Hamilton-Jacobi-
Bellman(HJB)-Isaacs equation of stochastic control and games thus allowing for
a unified analysis of these equations. This leads to an effective theory of
coupled forward-backward systems (forward McKean-Vlasov evolution and backward
HJB-Isaacs evolution) that are central to the modern theory of mean-field
games.

Mathematics Subject Classification (2010): 34A08, 35S15, 45G15

Key words: fractional McKean-Vlasov-type equations on manifolds, fractional
HJB-Isaacs equations on manifolds, fractional forward-backward systems on
manifolds, dual Banach triples, mild solutions, Caputo-Dzherbashyan fractional
derivative, smoothing and smoothness preserving operator semigroups

##  1 Introduction

We introduce a class of abstract nonlinear fractional pseudo-differential
equations in Banach spaces that includes both the Mc-Kean-Vlasov-type
equations describing nonlinear Markov processes and the Hamilton-Jacobi-
Bellman(HJB)-Isaacs equation of stochastic control and games thus allowing for
a unified analysis of these equations. Looking at these equations as evolving
in dual Banach triples allows us to recast directly the properties of one type
to the properties of another type leading to an effective theory of coupled
forward-backward systems (forward McKean-Vlasov evolution and backward HJB-
Isaacs evolution) that are central to the modern theory of mean-field games.
We are working with the mild solutions to the fractional nonlinear equations
that are based on the Zolotarev integral representation for the Mittag-Leffler
functions. The abstract setting developed allows us to include in our analysis
the related nonlinear fractional equations and forward-backward systems on
manifolds yielding results that are possibly new even for classical (not
fractional) equations. We obtain the well-posedness results for these
equations.

The present work is the continuation of the studies by the authors in [15, 16,
17].

We shall analyse the nonlinear Cauchy problems of the form

$\dot{b}(t)=Ab(t)+H(t,b(t),Db(t),\alpha),\quad b(a)=Y,\quad t\geq a,$ (1.1)

where $A$, $D_{1},\cdots,D_{n}$ are unbounded linear operators in a Banach
space $B$, $D=(D_{1},\cdots,D_{n})$, $\alpha$ is a parameter from another
Banach space $B^{par}$ and $H$ is a continuous mapping $\mathbf{R}\times
B\times B^{n}\times B^{par}\to B$, their fractional counterparts

$D^{\beta}_{a+*}b(t)=Ab(t)+H(t,b(t),Db(t),\alpha),\quad b(a)=Y\quad t\geq a,$
(1.2)

where $D^{\beta}_{a+*}$ is the Caputo-Dzherbashyan (CD) fractional derivative
of order $\beta\in(0,1)$,

$D^{\beta}_{a+*}b(t)=\frac{1}{\Gamma(-\beta)}\int_{0}^{t-a}\frac{b(t-z)-b(t)}{z^{1+\beta}}dz+\frac{b(t)-b(a)}{\Gamma(1-\beta)(t-a)^{\beta}},$
(1.3)

the anticipating versions of these equations (where $H$ depends additionally
on the future values of $b(s)$), and the forward-backward systems of coupled
equations of this type, which represent the main class of systems studied in
the modern theory of mean-field games.

######  Remark 1.1.

We gave the explicit formula for fractional derivative, which is a consequence
of its more standard definition as
$D^{\beta}_{a+*}b(t)=I_{a}^{1-\beta}(d/dt)b(t)$ via the fractional integral
$I^{\beta}$.

Our main examples concern the case when $B$ is a space of functions on
$\mathbf{R}^{d}$ and $D$ is the gradient (derivative) operator. Specifically,
the fractional Hamilton-Jacobi-Bellmann-Isaacs (HJB-Isaacs) equation of
controlled Markov processes (with an external parameter) is the equation of
the form

$D^{\beta}_{a+*}f(t,x)=Af(t,x)+H(t,x,f(t,x),\frac{\partial f}{\partial
x}(t,x),\alpha),$ (1.4)

for which the most natural Banach space is $B=C_{\infty}(\mathbf{R}^{d})$
($C_{\infty}(\mathbf{R}^{d})$ is the Banach space of continuous functions
$f:\mathbf{R}^{d}\to\mathbf{R}$ tending to zero at infinity equipped with the
sup-norm). The Hamiltonian $H$ arising from optimal control usually even does
not depend explicitly on $f$, just on its gradient, and it writes down as

$H(t,x,f,p,\alpha)=H(t,x,p,\alpha)=\sup_{u\in U}[J(t,x,u)+g(t,x,u)p],$ (1.5)

with some functions $J,g$, where $U$ is a compact set of controls (or with
$\inf\sup$ instead of just $\sup$ in case of Isaacs equations). For such $H$
the fractional equation (1.4) was derived in [18] as a Bellman equation for
optimal control of scaled limits of continuous time random walks. The
fractional version of the McKean-Vlasov type equations (describing nonlinear
Markov processes in the sense of [10]) are the quasi-linear equations of type

$D^{\beta}_{a+*}f(t,x)=A^{*}f(t,x)+\sum_{j=1}^{d}h_{j}(t,x,\\{f(t,.)\\},\alpha)\frac{\partial
f}{\partial x_{j}}(t,x),$ (1.6)

for which the most natural Banach space is $L_{1}(\mathbf{R}^{d})$ (or the
space of Borel measures on $\mathbf{R}^{d}$). In these equations $A$ is the
generator of a Feller process in $\mathbf{R}^{d}$ and $A^{*}$ is its dual
operators. While $H$ in (1.4) depends on the point-wise values of $f$, the
functions $h_{j}$ in (1.6) usually depend on some integrals of $f$. The
abstract framework of equations (1.2) allows one to treat these cases in a
unified way, as well as to include in the theory in a more or less
straightforward way important new cases, for instance, fractional HJB-Isaacs
or McKean-Vlasov-type equations on manifolds.

Fully analogous results of course hold for the backward versions of the Cauchy
problems above, namely for the problems

$\dot{b}(t)=-Ab(t)+H(t,b(t),Db(t),\alpha),\quad b(T)=Y,\quad t\leq T,$ (1.7)

and its fractional counterparts

$D^{\beta}_{T-*}b(t)=-Ab(t)+H(t,b(t),Db(t),\alpha),\quad b(T)=Y\quad t\leq T,$
(1.8)

where $D^{\beta}_{T-*}$ is the right Caputo-Dzherbashyan (CD) fractional
derivative of order $\beta\in(0,1)$:

$D^{\beta}_{T-*}b(t)=\frac{1}{\Gamma(-\beta)}\int_{0}^{T-t}\frac{b(t+z)-b(t)}{z^{1+\beta}}dt+\frac{b(t)-b(T)}{\Gamma(1-\beta)(T-t)^{\beta}}.$
(1.9)

The content of the paper is as follows. In the next two sections, we recall
preliminary material from the theory of the Mittag-Leffler functions and fixed
point principles. In Section 4 we present results on the well-posedness of
equations (1.1) and (1.2) in the sense of mild solutions. In Section 5 we
prove the local well-posedness for the anticipating versions of equations
(1.1) and (1.2) for sufficiently small $T-a$. In the next two sections 6 and 7
we present results on the well-posedness of the abstract coupled forward-
backward system and the fractional version of this forward-backward system.

In Section 8 we specify the abstract model for coupled forward-backward
systems and those fractional analogs looking at these equations as evolving in
dual Banach triples. We obtain the local well-posedness results for the
fractional coupled forward-backward system consisting of the coupled McKean
Vlasov (forward) and HJB (backward) equations for $t\in[a,T]$.

In Sections 9 and 10, the abstract setting developed in previous Section 8
allows us to prove the well-posedness results for the nonlinear fractional
equations and fractional coupled forward-backward systems on manifolds.

We shall work everywhere with mild solutions, but the standard arguments allow
one to get natural conditions that ensure that mild solutions are in fact
classical, see Theorem 6.1.3 of [13].

Note that fractional equations have become a popular subject of research due
to their wide applicability in various fields of natural sciences (see e.g.
[21, 22, 23]). We specially note the works [1] and [3], where the generalized
Euler-Lagrange equations and linear fractional differential equations are
considered. Numerical methods for fractional equations are presented in the
papers [4, 14]. The problem with two-sided fractional derivatives was analyzed
in [6]. Generalized fractional equations were considered in the works [7, 8,
12].

##  2 Preliminaries on the Mittag-Leffler functions

By $E_{\beta}(x)$ we denote the standard Mittag-Leffler function of index
$\beta$:

$E_{\beta}(x)=\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(\beta k+1)}.$

For our purpose the most convenient formula for the Mittag-Leffler function is
its integral representation (Zolotarev formula, or Zolotarev-Pollard formula)

$E_{\beta}(s)=\frac{1}{\beta}\int_{0}^{\infty}e^{sx}x^{-1-1/\beta}G_{\beta}(1,x^{-1/\beta})\,dx,$
(2.1)

where

$G_{\beta}(t,x)=\frac{1}{\pi}{Re}\int_{0}^{\infty}\exp\left\\{ipx-
tp^{\beta}e^{i\pi\beta/2}\right\\}dp$

is the heat kernel (solution with the Dirac initial condition) of the equation

$\frac{\partial G}{\partial t}(t,x)=-\frac{\partial^{\beta}}{\partial
x^{\beta}}G(t,x),$

or, in probabilistic terms, the transition probability density of the stable
Lévy subordinator of index $\beta$. The convenience of this formula is due to
the fact that it allows one to define $E_{\beta}(A)$ for an operator $A$
whenever $A$ generates a semigroup, so that $e^{At}$ is well defined.

From (2.1) it follows that

$E^{\prime}_{\beta}(s)=\frac{1}{\beta}\int_{0}^{\infty}e^{sx}x^{-1/\beta}G_{\beta}(1,x^{-1/\beta})\,dx,$
(2.2)

so that the integral on r.h.s. is finite.

We also need the well known formula for the Mellin transform of $G_{\beta}$:

$\int_{0}^{\infty}x^{-\omega}G_{\beta}(1,x^{-1/\beta})\,dx=\frac{\Gamma(1-\omega+1/\beta)}{\beta\Gamma(\beta-\beta\omega+1)},$
(2.3)

valid for $\omega<1+1/\beta$, see proof e.g. in Proposition 8.1.1 of [13].

##  3 Preliminaries on the fixed-point principle for integral curves

For a Banach space $B$ and $\tau<t$ we denote $C([\tau,t],B)$ the Banach space
of continuous functions $f:[\tau,t]\to B$ with the norm

$\|f\|_{C([\tau,t],B)}=\sup_{s\in[\tau,t]}\|f(s)\|_{B},$

and $C_{Y}([\tau,t],B)$ its closed subset consisting of functions $f$ such
that $f(\tau)=Y$, which is a complete metric space under the induced topology.

For a closed convex subset $M$ of $B$, $C_{Y}([\tau,t],M)$ denotes a convex
subset of $C_{Y}([\tau,t],B)$ of functions with values in $M$.

The following result is Theorem 2.1.3 from [13]. It is a version of the fixed-
point principle, specifically tailored to be used for nonlinear diffusions and
fractional equations.

######  Theorem 3.1.

Suppose that for any $Y\in M$, $\alpha\in B_{1}$, with $B_{1}$ another Banach
space, a mapping $\Phi_{Y,\alpha}:C([\tau,T],M)\to C_{Y}([\tau,T],M)$ is given
with some $T>\tau$ such that for any $t$ the restriction of
$\Phi_{Y,\alpha}(\mu_{.})$ on $[\tau,t]$ depends only on the restriction of
the function $\mu_{s}$ on $[\tau,t]$. Moreover,

$\displaystyle\|[\Phi_{Y,\alpha}(\mu^{1}_{.})](t)-[\Phi_{Y,\alpha}(\mu^{2}_{.})](t)\|\leq
L(Y)\int_{\tau}^{t}(t-s)^{-\omega}\|\mu^{1}_{.}-\mu^{2}_{.}\|_{C([\tau,s],B)}\,ds,$
(3.1)
$\displaystyle\|[\Phi_{Y_{1},\alpha_{1}}(\mu_{.})](t)-[\Phi_{Y_{2},\alpha_{2}}(\mu_{.})](t)\|\leq\varkappa\|Y_{1}-Y_{2}\|+\varkappa_{1}\|\alpha_{1}-\alpha_{2}\|,$

for all $t\in[\tau,T]$, $\mu^{1},\mu^{2}\in C([\tau,T],M)$,
$\alpha_{1},\alpha_{2}\in B_{1}$, some constants $\varkappa,\varkappa_{1}\geq
0$, $\omega\in[0,1)$, and a continuous function $L$ on $M$.

Then for any $Y\in M$, $\alpha\in B_{1}$ the mapping $\Phi_{Y,\alpha}$ has a
unique fixed point $\mu_{t,\tau}(Y,\alpha)$ in $C_{Y}([\tau,T],M)$. Moreover,
if $\omega>0$, then for all $t\in[\tau,T]$,

$\|\mu_{t,\tau}(Y,\alpha)-Y\|\leq
E_{1-\omega}(L(Y)\Gamma(1-\omega)(t-\tau)^{1-\omega})\|[\Phi_{Y,\alpha}(Y)](t)-Y\|,$
(3.2)

and the fixed points $\mu_{t,\tau}(Y_{1},\alpha_{1})$ and
$\mu_{t,\tau}(Y_{2},\alpha_{2})$ with different initial data $Y_{1},Y_{2}$ and
parameters $\alpha_{1},\alpha_{2}$ enjoy the estimate (for any $j=1,2$)

$\|\mu_{t,\tau}(Y_{1},\alpha_{1})-\mu_{t,\tau}(Y_{2},\alpha_{2})\|\leq(\varkappa\|Y_{1}-Y_{2}\|+\varkappa_{1}\|\alpha_{1}-\alpha_{2}\|)E_{1-\omega}(L(Y_{j})\Gamma(1-\omega)(t-\tau)^{1-\omega}).$
(3.3)

If $\omega=0$, these estimates are simplified to

$\|\mu_{t,\tau}(Y,\alpha)-Y\|\leq
e^{(t-\tau)L(Y)}\|[\Phi_{Y,\alpha}(Y)](t)-Y\|,$ (3.4)
$\|\mu_{t,\tau}(Y_{1},\alpha_{1})-\mu_{t,\tau}(Y_{2},\alpha_{2})\|\leq(\varkappa\|Y_{1}-Y_{2}\|+\varkappa_{1}\|\alpha_{1}-\alpha_{2}\|)\exp\\{(t-\tau)\min(L(Y_{1}),L(Y_{2}))\\}.$
(3.5)

##  4 Abstract fractional McKean-Vlasov and HJB equations

For two Banach spaces $B,C$ we denote by $\mathcal{L}(B,C)$ the Banach space
of bounded linear operators $B\to C$ with the usual operator norm denoted
$\|.\|_{B\to C}$.

The sequences of embedded Banach spaces $B_{2}\subset B_{1}\subset B$ with the
norms denoted $\|.\|_{2},\|.\|_{1},\|.\|$ respectively, will be referred to as
the Banach triple (of embedded spaces) or a Banach tower of order $3$, if the
norms are ordered, $\|.\|_{2}\geq\|.\|_{1}\geq\|.\|$, and $B_{2}$ is dense in
$B_{1}$ in the topology of $B_{1}$ while $B_{1}$ is dense in $B$ in the
topology of $B$. The following setting will play the key role in this paper.

Conditions (A):

(i) Let $B_{2}\subset B_{1}\subset B$ be the Banach triple, with the norms
denoted $\|.\|_{2},\|.\|_{1},\|.\|$ respectively, and let

$D_{i}\in\mathcal{L}(B_{1},B)\cap\mathcal{L}(B_{2},B_{1}),\quad i=1,\cdots,n.$

Without loss of generality we assume that norms of all $D_{j}$ are bounded by
$1$ in both $\mathcal{L}(B_{1},B)$ and $\mathcal{L}(B_{2},B_{1})$ (which is
usually the case in applications below).

(ii) Let $A\in\mathcal{L}(B_{2},B)$ and let $A$ generate a strongly continuous
semigroup $e^{At}$ in both $B$ and $B_{1}$, so that

$\|e^{At}\|_{B\to B}\leq Me^{mt},\quad\|e^{At}\|_{B_{1}\to B_{1}}\leq
M_{1}e^{tm_{1}},$ (4.1)

with some nonnegative constants $M,m,M_{1},m_{1}$, and $B_{2}$ is an invariant
core for this semigroup in $B$.

(iii) Let $B^{par}$ be another Banach space (of parameters) with the norm
denoted $\|.\|_{par}$ and $H:\mathbf{R}\times B\times B^{n}\times B^{par}\to
B$ be a continuous mapping, which is Lipschitz in the sense that

$\|H(t,b_{0},b_{1},\cdots,b_{n},\alpha)-H(t,\tilde{b}_{0},\tilde{b}_{1},\cdots,\tilde{b}_{n},\alpha)\|$
$\leq
L_{H}\sum_{j=0}^{n}\|b_{j}-\tilde{b}_{j}\|(1+L^{\prime}_{H}\sum_{j=1}^{n}\|b_{j}\|),$
(4.2)
$\|H(t,b_{0},b_{1},\cdots,b_{n},\alpha)-H(t,b_{0},b_{1},\cdots,b_{n},\tilde{\alpha})\|\leq
L_{H}^{par}\|\alpha-\tilde{\alpha}\|_{par}(1+\sum_{j=1}^{n}\|b_{j}\|),$ (4.3)

and is of linear growth

$\|H(t,b_{0},b_{1},\cdots,b_{n},\alpha)\|\leq L_{H}\sum_{j=1}^{n}\|b_{j}\|,$
(4.4)

with some constants $L_{H},L^{\prime}_{H},L_{H}^{par}$.

######  Remark 4.1.

For the classical HJB equations estimate (4.2) holds with $L^{\prime}_{H}=0$,
in which case (4.4) follows from (4.2) and (4.3) (with $L_{H}$ that may depend
on $\alpha$). However, for McKean-Vlasov-type equations, the linear growth of
the Lipschitz constant in (4.2) is not avoidable.

An important assumption in our analysis is the following smoothing property of
the semigroup $e^{At}$: for $t>0$ it takes $B$ to $B_{1}$ and

$\|e^{At}\|_{B\to B_{1}}\leq\varkappa t^{-\omega},\quad t\in(0,1],$ (4.5)

with some constants $\varkappa>0$ and $\omega\in(0,1)$. Sometimes a similar
condition for the pair $B_{1},B_{2}$ is used:

$\|e^{At}\|_{B_{1}\to B_{2}}\leq\varkappa_{1}t^{-\omega},\quad t\in(0,1].$
(4.6)

######  Remark 4.2.

For pseudo-differential operators $A$ (including the generators of Feller
semigroups) the deeper smoothing property (4.6) can be derived from (4.5) and
the smoothness of the symbol of $A$, see Theorem 5.15.1 in [13].

The so-called mild version of the Cauchy problem (1.1) is the integral
equation

$b(t)=e^{A(t-a)}Y+\int_{a}^{t}e^{A(t-s)}H(s,b(s),Db(s),\alpha)\,ds,\quad t\geq
a.$ (4.7)

It is well known (see e.g. [13]) and easy to see that if $b(t)$ solves
equation (1.1), then it solves also equation (4.7), so that the uniqueness for
(4.7) implies the uniqueness for (1.1).

The following theorem on the well-posedness of the equations (4.7) is valid.

######  Theorem 4.1.

Let conditions (A) and smoothing property (4.5) hold. Then equation (4.7) is
well posed in $B_{1}$, that is, for any $Y\in B_{1}$, $\alpha\in B^{par}$
there exists its unique global solution $b(t)=b(t;Y,\alpha)\in B_{1}$, which
depends Lipschitz continuously on the initial data $Y$ and the parameter
$\alpha$. In particular,

$\sup_{t\in[a,T]}\|b(t;Y,\alpha)-b(t;\tilde{Y},\tilde{\alpha})\|_{1}\leq
K\left(\|\alpha-\tilde{\alpha}\|_{par}(1+\|Y\|_{1})+\|Y-\tilde{Y}\|_{1}\right),$
(4.8)

with constant $K$ depending on $t-a$ and all constants entering the
assumptions of the theorem.

###### Proof.

Solutions to (4.7) are fixed points of the mapping

$[\Phi_{Y,\alpha}(b(.))](t)=e^{A(t-a)}Y+\int_{a}^{t}e^{A(t-s)}H(s,b(s),Db(s),\alpha)\,ds$
(4.9)

acting in $C_{Y}([a,T],B_{1})$ for any $T>a$. The fact that it takes this
space to itself follows directly from the assumptions of the Theorem.

Assume first that $L^{\prime}_{H}=0$ in (4.2). Then it follows that

$\|[\Phi_{Y_{1},\alpha}(b(.))](t)-[\Phi_{Y_{2},\alpha}(b(.))](t)\|_{1}\leq
M_{1}e^{m_{1}t}\|Y_{1}-Y_{2}\|_{1},$ (4.10)

and

$\|[\Phi_{Y,\alpha}(b^{1}(.))](t)-[\Phi_{Y,\alpha}(b^{2}(.))](t)\|_{1}$
$\leq\varkappa
L_{H}(T-a)^{\omega}M_{1}e^{m_{1}(T-a)}\int_{a}^{t}(t-s)^{-\omega}\left(\|b^{1}(s)-b^{2}(s)\|+\sum_{j=1}^{n}\|D_{j}b^{1}(s)-D_{j}b^{2}(s)\|\right)\,ds$
$\leq\varkappa
L_{H}(n+1)(T-a)^{\omega}M_{1}e^{m_{1}(T-a)}\int_{a}^{t}(t-s)^{-\omega}\|b^{1}(s)-b^{2}(s)\|_{1}\,ds.$
(4.11)

Note that the coefficient $(T-a)^{\omega}M_{1}e^{m_{1}(T-a)}$ appears here
because smoothing (4.5) is assumed only for $t<1$. Due to these inequalities,
the well-posedness for any $\alpha$ follows from Theorem 3.1. Moreover, from
(4.3), we find that

$\|[\Phi_{Y,\alpha}(b(.))](t)-[\Phi_{Y,\tilde{\alpha}}(b(.))](t)\|_{1}\leq\tilde{K}\|\alpha-\tilde{\alpha}\|_{par}(1+\|b(.)\|_{C([\tau,T],B_{1})}),$

and therefore, if $b$ is a fixed point of $\Phi$, its growth is bounded by
(3.4), and thus

$\|[\Phi_{Y,\alpha}(b(.))](t)-[\Phi_{Y,\tilde{\alpha}}(b(.))](t)\|_{1}\leq
K\|\alpha-\tilde{\alpha}\|_{par}(1+\|Y\|_{1}),$

with some constants $\tilde{K},K$ depending on the constants entering the
assumptions of the theorem. Hence Theorem 3.1 (with $\varkappa_{1}$ from (3.1)
depending on $\|Y\|_{1}$) again applies to give (4.8).

If $L^{\prime}_{H}\neq 0$ in (4.2), an additional preliminary step is needed
in the proof. Namely, one has to show that all iterations are uniformly
bounded in $B_{1}$. This follows from the assumptions of linear growth (4.4)
and then applies the previous argument. ∎

######  Remark 4.3.

Of course, Theorem 3.1 supplies also explicit estimates for the constant $K$
and the growth of solutions in time.

To treat equation (1.2) we recall that its mild form is the integral equation

$b(t)=E_{\beta}(A(t-a)^{\beta})Y+\beta\int_{a}^{t}(t-s)^{\beta-1}E_{\beta}^{\prime}(A(t-s)^{\beta})H(s,b(s),Db(s),\alpha)\,ds,$
(4.12)

where $E_{\beta}(A)$ is defined by (2.1). Thus more explicitly this equation
writes down as

$b(t)=\frac{1}{\beta}\int_{0}^{\infty}e^{A(t-a)^{\beta}x}Yx^{-1-1/\beta}G_{\beta}(1,x^{-1/\beta})\,dx$
$+\int_{a}^{t}(t-s)^{\beta-1}\int_{0}^{\infty}e^{A(t-s)^{\beta}x}x^{-1/\beta}G_{\beta}(1,x^{-1/\beta})\,dxH(s,b(s),Db(s),\alpha)\,ds.$
(4.13)

Again one proves (see Theorem 8.2.1 of [13]) that any solution of problem
(1.2) solves (4.12).

######  Theorem 4.2.

Let conditions (A) and smoothing (4.5) hold. Then equation (4.12) is well
posed in $B_{1}$, that is, for any $Y\in B_{1}$, $\alpha\in B^{par}$ there
exists its unique global solution $b(t)=b(t;Y,\alpha)\in B_{1}$, which depends
Lipschitz continuously on the initial data $Y$ and parameter $\alpha$ so that
(4.8) holds.

###### Proof.

For simplicity, let us discuss again only the case with $L^{\prime}_{H}=0$ in
(4.2). Solutions to (4.7) are fixed points of the mapping

$[\Phi_{Y,\alpha}(b(.))](t)=E_{\beta}(A(t-a)^{\beta})Y+\beta\int_{a}^{t}(t-s)^{\beta-1}E_{\beta}^{\prime}(A(t-s)^{\beta})H(s,b(s),Db(s),\alpha)\,ds$

acting in $C_{Y}([a,T],B_{1})$ for any $T>a$. We have

$\|[\Phi_{Y_{1},\alpha}(b(.))](t)-[\Phi_{Y_{2},\alpha}(b(.))](t)\|_{1}\leq
M_{1}E_{\beta}(m_{1}(t-a)^{\beta})\|Y_{1}-Y_{2}\|_{1},$ (4.14)

which follows from (4.1) and (2.1). Next,

$[\Phi_{Y,\alpha}(b^{1}(.))](t)-[\Phi_{Y,\alpha}(b^{2}(.))](t)=\int_{a}^{t}(t-s)^{\beta-1}ds$
$\times\int_{0}^{\infty}e^{A(t-s)^{\beta}x}x^{-1/\beta}G_{\beta}(1,x^{-1/\beta})dx[H(s,b^{1}(s),Db^{1}(s),\alpha)-H(s,b^{2}(s),Db^{2}(s),\alpha)]\,ds.$

Decomposing the double integral in two parts over the sets $(t-a)^{\beta}x\leq
1$ and its complement, we use estimate (4.5) in the first integral and the
second estimate from (4.1) in the second integral yielding

$\|[\Phi_{Y,\alpha}(b^{1}(.))](t)-[\Phi_{Y,\alpha}(b^{2}(.))](t)\|_{1}$
$\leq\varkappa
L_{H}(n+1)\int_{a}^{t}(t-s)^{\beta-1}(t-s)^{-\omega\beta}\int_{0}^{\infty}x^{-\omega-1/\beta}G_{\beta}(1,x^{-1/\beta})\,dx\|b^{1}(s)-b^{2}(s)\|_{1}\,ds$
$+\varkappa
L_{H}M_{1}(n+1)\int_{a}^{t}(t-s)^{\beta-1}\int_{0}^{\infty}e^{m_{1}(t-s)^{\beta}x}x^{-1/\beta}G_{\beta}(1,x^{-1/\beta})\,dx\|b^{1}(s)-b^{2}(s)\|_{1}\,ds.$

Using (2.3) to estimate the first term and (2.2) to estimate the second term
yields

$\|[\Phi_{Y,\alpha}(b^{1}(.))](t)-[\Phi_{Y,\alpha}(b^{2}(.))](t)\|_{1}$ $\leq
L_{H}(n+1)(\varkappa+M_{1})C(m_{1},\beta,T-a)\int_{a}^{t}(t-s)^{-[1-\beta(1-\omega)]}\|b^{1}(s)-b^{2}(s)\|_{1}\,ds,$
(4.15)

with a constant $C$ depending on $m_{1},\beta,T-a$. Due to these inequalities,
and similar inequality expressing the Lipschitz dependence of $\Phi$ on
$\alpha$, the claim follows again from Theorem 3.1. ∎

######  Remark 4.4.

Notice that $E_{1}(x)=e^{x}$ and thus $E^{\prime}_{1}(x)=e^{x}$. Hence the
fractional mild equation (4.12) turns to the classical mild equation (4.7) as
$\beta\to 1$. Hence the discussion of the case of classical differential
equations can be considered as included in the fractional theory. In later
sections, we shall sometimes talk about just fractional equations having in
mind that the classical case is recovered automatically by setting $\beta=1$.

######  Remark 4.5.

All the results above have their straightforward counterparts for the backward
problems under the same exact conditions. The only difference is that, instead
of the sets $C_{Y}([\tau,t],B)$ of functions with a fixed left point, in the
backward setting one is working with the sets $C^{b}_{Z}([\tau,T],B)$ of
functions with a fixed right point: $f(T)=Z$.

Let us comment on how the results above are applied to HJB and McKean-Vlasov
equations (1.4) and (1.6). For these cases, $B$ is a space of functions on
$\mathbf{R}^{d}$, $A$ is the generator of a Feller process in $R^{d}$ and $D$
is the gradient (derivative) operator. Specifically, for HJB equation the
natural Banach triple is $C^{2}_{\infty}(\mathbf{R}^{d})\subset
C^{1}_{\infty}(\mathbf{R}^{d})\subset C_{\infty}(\mathbf{R}^{d})$, and for the
McKean-Vlasov equations a possible triple is $W_{2}(\mathbf{R}^{d})\subset
W_{1}(\mathbf{R}^{d})\subset L_{1}(\mathbf{R}^{d})$. Here
$C_{\infty}(\mathbf{R}^{d})$ denotes the space of continuous functions on
$\mathbf{R}^{d}$ tending to zero at infinity, $C^{j}_{\infty}(\mathbf{R}^{d})$
their subsets of functions having derivatives of order up to $j$ in
$C_{\infty}(\mathbf{R}^{d})$, and $W_{j}(R^{d})$ denote the Sobolev spaces of
functions with partial derivatives (understood in the sense of generalized
functions) of order up to $j$ in $L_{1}(\mathbf{R}^{d})$, equipped with the
integral (Sobolev) norms

$\|f\|_{L_{1}(R^{d})}=\int|f(x)|\,dx,\quad\|f\|_{W_{1}(R^{d})}=\|f\|_{L_{1}(R^{d})}+\sum_{j=1}^{d}\int\left|\frac{\partial
f}{\partial x_{j}}\right|dx,$
$\|f\|_{W_{2}(R^{d})}=\|f\|_{W_{1}(R^{d})}+\sum_{i\leq
j}\int\left|\frac{\partial^{2}f}{\partial x_{j}\partial x_{i}}\right|dx.$

The proof of the smoothing properties of $A$ is usually based on the
properties of the Green functions of operator $A$ (transition probabilities of
the processes generated by $A$). For instance, (4.5) and (4.6) are known to
hold with $\omega=1/2$ for non-degenerate diffusion operators $A$ with smooth
enough coefficients. When $A=-|\Delta|^{\alpha}$ with $\alpha\in(1,2)$, or
more generally $A=-a(x)|\Delta|^{\alpha}$ with a smooth positive function $a$
bounded from above and below, or even more generally when $A$ is a pseudo-
differential operator that generates a non-degenerate stable-like process with
a symmetric spectral measure, that is

$A=\int_{S^{d-1}}|(\nabla,s)|^{\alpha}\mu(s)\,ds,$

with a positive smooth function $\mu$ on $(d-1)$-dimensional sphere $S^{d-1}$,
bounded from above and below, the semigroup generated by $A$ satisfies (4.5),
(4.6) and (4.1), with $\omega=1/\alpha$, as shown in [9] (see also books [11]
and [13]). Similar properties are also being held for various mixtures of
diffusions and stable processes perturbed by pure jump processes. As shown in
Theorem 8.1 below, from the properties of $A$ one can derive analogous
properties for $A^{*}$ by duality arguments. Thus for these $A$ the required
assumptions on $A$ and $A^{*}$ from (1.4) and (1.6) hold. We also refer to
Section 5.15 of book [13], where it is shown, how one can derive (4.6) from a
more simple estimate (4.5) and additional smoothness. For Hamiltonian
functions (1.5) assumptions (B) are often difficult to check, and therefore
the assumptions (C) were introduced. The theory of fractional HJB equations
was initially developed in [19], with more detail given in [13].

######  Remark 4.6.

From an abstract point of view the spaces $C_{\infty}(\mathbf{R}^{d})$ and
$L_{1}(\mathbf{R}^{d})$ are the basic examples of abstract $AL$ and $AM$
spaces (see [24]).

##  5 Anticipating equations

Anticipating versions of equations (1.1) and (1.2) can be stated as the
problems

$\dot{b}(t)=Ab(t)+H(t,b(t),Db(t),u(b(.))),\quad b(a)=Y,\quad t\in[a,T],$ (5.1)
$D^{\beta}_{a+*}b(t)=Ab(t)+H(t,b(t),Db(t),u(t,b(.))),\quad b(a)=Y\quad t\geq
a,$ (5.2)

where $A$, $D_{1},\cdots,D_{n}$, $H$ are as in (1.1) and (1.2), and $u$ is a
continuous mapping $\mathbf{R}\times C([a,T],B_{1})\to B^{par}$, which is
Lipschitz in the second argument, so that

$\|u(t,b(.))-u(t,\tilde{b}(.))\|_{par}\leq
L_{u}\|b(.)-\tilde{b}(.)\|_{C([a,T],B_{1})}.$ (5.3)

Usually, in applications $u(t,.)$ depends actually only on the future
$b(s):s\in[t,T]$, but this additional condition does not simplify analysis.

For these problems uniqueness usually does not hold globally (that is for
large $T$), only existence can be proved under rather general assumptions (see
e.g. [20] for equations of type (5.1)). We shall prove here the local well-
posedness, that is, for sufficiently small $T-a$. Of course, we again work
with mild solutions, relying on the fact that any solution of (5.1) is a fixed
point of the mapping

$[\Phi_{Y}(b(.))](t)=e^{A(t-a)}Y+\int_{a}^{t}e^{A(t-s)}H(s,b(s),Db(s),u(s,b(.)))\,ds,$
(5.4)

and any solution of (5.2) is a fixed point of the mapping

$[\Phi_{Y}(b(.))](t)=E_{\beta}(A(t-a)^{\beta})Y+\beta\int_{a}^{t}(t-s)^{\beta-1}E_{\beta}^{\prime}(A(t-s)^{\beta})H(s,b(s),Db(s),u(s,b(.)))\,ds.$
(5.5)

Due to the anticipating dependence of $u$ on $b$, Theorem 3.1 is not
applicable. We shall use just the standard Banach contraction principle.

######  Theorem 5.1.

Let conditions (A), smoothing property (4.5) and Lipschitz estimate (5.3)
hold. Then, for any $r>0$, there exists $T_{0}$ such that equations (5.1) and
(5.2) have unique mild solutions $b(t)=b(t;Y)$ (that is, they are fixed points
of (5.4) and (5.5) respectively) for all $T<T_{0}$ and $Y$ such that
$\|Y\|\leq r$, and this solution depends Lipschitz continuously on $Y$ inside
the ball $\|Y\|\leq r$.

###### Proof.

Due to formula (2.1) expressing Mittag-Leffler functions in terms of
exponents, the proof is essentially the same for both equations. So let us
discuss only equation (5.1). There are two steps. Firstly, due to the linear
growth condition (4.4), one derives for $\|Y\|_{1}\leq r$ the estimate

$\|[\Phi_{Y}(b(.))](t)\|_{1}\leq
M_{1}e^{m_{1}T}r+\epsilon\|b(.)\|_{C_{Y}([a,T],B_{1})},$

where $\epsilon$ depends on $T_{0}$, so that one can choose $T_{0}$ in such a
way that $\epsilon<1$. In this case if $\sup_{t}\|b(.)\|_{1}\leq R$, then

$\|[\Phi_{Y}(b(.))](t)\|_{1}\leq M_{1}e^{m_{1}T}r+\epsilon R\leq R,$

whenever

$R>\frac{M_{1}e^{m_{1}T}r}{1-\epsilon}.$

Thus for any such $R$, $\Phi_{Y}$ takes $C_{Y}([a,T],B_{1}^{R})$ to itself,
where $B_{1}^{R}$ is the ball in $B_{1}$ of radius $R$ centred at $0$. Next,
from the estimate

$\|[\Phi_{Y}(b^{1}(.))](t)-\Phi_{Y}(b^{2}(.))(t)\|_{1}\leq\|b^{1}(.)-b^{2}(.)\|_{C_{Y}([a,T],B_{1})}(T-a)e^{m_{1}(T-a)}[k_{1}+k_{2}(\|Y\|_{1}+R)],$

where $k_{1},k_{2}$ do not depend on $R$ and $Y$, it follows that $\Phi_{Y}$
is a contraction in the complete metric spaces $C_{Y}([a,T],B_{1}^{R})$ for
sufficiently small $T-a$. ∎

##  6 Forward-backward systems

The well-posedness result of this section is an abstract version of the
results obtained in [20] (slightly extended in Chapter 6 of book [13]). This
abstract presentation not only simplifies the exposition, but it is
specifically designed for a more or less straightforward extension to the
fractional case, as given in the next section, and which is our main concern
here.

Paper [20] contains also a global existence result (without uniqueness) that
can be recast in the present abstract setting, but we are not giving detail
here.

To introduce our coupled forward backward system we need to introduce
conditions (A) both for the forward and the backward parts of the system plus
the coupling mechanism (interpreted as control in applications) including an
appropriate setting of the parameter space to support this mechanism. That is,
the forward-backward version of conditions (A) reads as follows.

Conditions (AFB):

(i) Let $B_{2}\subset B_{1}\subset B$ and $B_{2b}\subset B_{1b}\subset B_{b}$
be two Banach triples (index ’b’ arising from ’backward’), with the norms in
the second triple denoted $\|.\|_{2b}$, $\|.\|_{1b}$ and $\|.\|_{b}$. Let
$D=(D_{1},\cdots,D_{n})$ and $D^{b}=(D_{1b},\cdots,D_{nb})$ be operators
satisfying conditions (A) (i) for the triples $B_{2}\subset B_{1}\subset B$
and $B_{2b}\subset B_{1b}\subset B_{b}$, respectively.

(ii) Let $A$ and $A^{b}$ be operators with properties described in condition
(A)(ii) for the triples $B_{2}\subset B_{1}\subset B$ and $B_{2b}\subset
B_{1b}\subset B_{b}$, respectively. The corresponding constants governing the
growth of semigroups for the second triple will be denoted
$M_{b},m_{b},M_{1b},m_{1b}$.

(iii) Let $B^{con}$ be another Banach space (of control functions) with the
norm denoted $\|.\|_{con}$. Let $H:\mathbf{R}\times B\times B^{n}\times
B^{con}\to B$ be a continuous mapping, which is Lipschitz in the sense that

$\|H(t,b_{0},b_{1},\cdots,b_{n},u)-H(t,\tilde{b}_{0},\tilde{b}_{1},\cdots,\tilde{b}_{n},u)\|\leq
L_{H}\sum_{j=0}^{n}\|b_{j}-\tilde{b}_{j}\|(1+L^{\prime}_{H}\sum_{j=1}^{n}\|b_{j}\|),$
(6.1)
$\|H(t,b_{0},b_{1},\cdots,b_{n},u)-H(t,b_{0},b_{1},\cdots,b_{n},\tilde{u})\|\leq
L_{H}^{par}\|u-\tilde{u}\|_{con}(1+\sum_{j=1}^{n}\|b_{j}\|).$ (6.2)

Let $H^{b}:\mathbf{R}\times B_{b}\times(B_{b})^{n}\times C([a,T],B)\to B_{b}$
be a continuous mapping, which is Lipschitz in the sense that

$\|H^{b}(t,f_{0},f_{1},\cdots,f_{n},b(.))-H^{b}(t,\tilde{f}_{0},\tilde{f}_{1},\cdots,\tilde{f}_{n},b(.))\|_{b}\leq
L^{b}_{H}\sum_{j=0}^{n}\|f_{j}-\tilde{f}_{j}\|_{b}(1+(L_{H}^{b})^{\prime}\sum_{j=1}^{n}\|b_{j}\|_{b}),$
(6.3)
$\|H^{b}(t,f_{0},f_{1},\cdots,f_{n},b(.))-H^{b}(t,f_{0},f_{1},\cdots,f_{n},\tilde{b}(.))\|\leq
L_{H}^{b}\|b-\tilde{b}\|_{C([a,T],B)}(1+\sum_{j=0}^{n}\|f_{j}\|_{b}).$ (6.4)

Finally let both $H$ and $H^{b}$ be of linear growth (in the sense of equation
(4.4)).

(iv) Let $u:\mathbf{R}\times B_{b}\times(B_{b})^{n}\to B^{con}$ be a
continuous function, which is Lipschitz continuous in the sense that

$\|u(t,f_{0},f_{1},\cdots,f_{n})-u(t,\tilde{f}_{0},\tilde{f}_{1},\cdots,\tilde{f}_{n})\|\leq
L_{u}\sum_{j=0}^{n}\|f_{j}-\tilde{f}_{j}\|_{b},$ (6.5)

with a constant $L_{u}$.

The forward-backward system we are analysing here is of the form

$\left\\{\begin{aligned}
&\dot{b}(t)=Ab(t)+H(t,b(t),Db(t),u(t,f(t),Df(t))),\quad b(a)=Y,\quad
t\in[a,T],\\\ &\dot{f}(t)=-A^{b}f(t)+H^{b}(t,f(t),D^{b}f(t),b_{\geq t}),\quad
f(T)=Z,\quad t\in[a,T].\end{aligned}\right.$ (6.6)

The notation $b_{\geq t}$ indicates the assumption that $H(t,.)$ depends only
on the future values $\\{b(s),s\in[t,T]\\}$ of curves $b(.)\in C([a,T],B)$.

Our approach for the analysis of system (6.6) is based on its reduction to a
single anticipating equation of type (5.1). Namely, by Theorem 4.1 (and Remark
4.5) for a given curve $b(.)\in C_{Y}([a,T],B)$ and $Z\in B_{1b}$ there exists
a unique mild solution $f(t)=f(t;Z,b(.))\in C([a,T],B_{1b})$ of the second
equation of system (6.6). Substituting this solution into the first equation
of system (6.6) we get the equation

$\dot{b}(t)=Ab(t)+H(t,b(t),Db(t),u(t,f(t;Z,b(.)),Df(t;Z,b(.)))),\quad
t\in[a,T].$ (6.7)

This equation is of type (5.1).

######  Theorem 6.1.

Let conditions (AFB) and smoothing property (4.5) for $A$ and $A^{b}$ hold.
Then, for any $Y\in B_{1}$, $Z\in B_{1b}$ there exists $T_{0}$ such that the
forward-backward system (6.6) has unique mild solution $(b(t),f(t))$ for all
$T<T_{0}$, which depends locally Lipschitz on $Y$, $Z$, that is, when are $Y$
and $Z$ are taken from bounded sets.

###### Proof.

The discussion above shows that the statement is reduced to the well-posedness
of equation (6.7). To apply Theorem 5.1 we just have to check the Lipschitz
estimate (5.3), which in the present case has the form

$\|u(t,f(t;Z,b(.)),Df(t;Z,b(.)))-u(t,f(t;Z,\tilde{b}(.)),Df(t;Z,\tilde{b}(.)))\|_{con}$
(6.8) $\leq\|b(.)-\tilde{b}(.)\|_{C([a,T],B_{1})}.$

But this holds, because the function $u$ is a Lipschitz continuous function of
its arguments (by (6.5)) and $f$ and $Df$ are Lipschitz functions of $b(.)$ by
Theorem 4.1. Let us stress that this application was the main reason for us to
get the continuous dependence of the solutions of the HJB and McKean-Vlasov
equations in a ’deeper’ space $B_{1}$ (not just in $B$). ∎

##  7 Fractional forward-backward systems

The fractional analogue of system (6.6) is ,of course, the system

$\left\\{\begin{aligned}
&D_{a+*}b(t)=Ab(t)+H(t,b(t),Db(t),u(t,f(t),Df(t))),\quad b(a)=Y,\quad
t\in[a,T],\\\ &D_{T-*}f(t)=-A^{b}f(t)+H^{b}(t,f(t),D^{b}f(t),b_{\geq t}),\quad
f(T)=Z,\quad t\in[a,T].\end{aligned}\right.$ (7.1)

Under the same assumptions (AFB) we can conclude from Theorem 4.2 that for a
given curve $b(.)\in C_{Y}([a,T],B)$ and $Z\in B_{1b}$ there exists a unique
mild solution $f(t)=f(t;Z,b(.))\in C([a,T],B_{1b})$ of the second equation of
system (7.1). Substituting this solution into the first equation of system
(7.1) we get the equation

$D_{a+*}b(t)=Ab(t)+H(t,b(t),Db(t),u(t,f(t;Z,b(.)),Df(t;Z,b(.)))),\quad
t\in[a,T],$ (7.2)

which is of type (5.2).

The following result is proved by the same exact argument as Theorem 6.1:

######  Theorem 7.1.

Let conditions (AFB) and smoothing property (4.5) for $A$ and $A^{b}$ hold.
Then, for any $Y\in B_{1}$, $Z\in B_{1b}$ there exists $T_{0}$ such that the
forward-backward system (7.1) has unique mild solution $(b(t),f(t))$ for all
$T<T_{0}$, which depends locally Lipschitz on $Y$ and $Z$.

##  8 Dual Banach triples and related nonlinear equations

In applications the two Banach triples participating in (AFB) are usually
linked by duality, and operators $A,A^{b}$ are dual, which allows one to
derive the required properties of $A^{b}$ from the corresponding properties of
$A$.

It is convenient to introduce the corresponding setting in an abstract way.
Following [13], we shall call a pair of Banach space $(B,C)$ a dual pair, if
each of these spaces is a closed subset of the dual of the respective other
space that separates the points of the latter. Of course, if $B^{\prime}$ is
the Banach dual of $B$, then both $(B,B^{\prime})$ and $(B^{\prime},B)$ are
dual pairs. But our symmetric notion includes also the pairs like
$(C_{\infty}(\mathbf{R}^{d}),L_{1}(\mathbf{R}^{d}))$, which are crucial for
our present discussion. The duality is a bilinear form $(B,C)\to\mathbf{R}$,
which will be always denoted by $(b,c)$, $b\in B,c\in C$. Recall that the
requirement that $B$ is a subset of the dual to $C$ means the following
formula for the norm:

$\|b\|_{B}=\sup_{\|c\|\leq 1}|(b,c)|.$

Let us say that the Banach triple $B_{2}\subset B_{1}\subset B$ is generated
by the vector-valued operator $D=(D_{1},\cdots,D_{n})$ if

$\|b\|_{1}=\|b\|+\sum_{j=1}^{n}\|D_{j}b\|,\quad\|b\|_{2}=\|b\|_{1}+\sum_{i\leq
j}\|D_{i}D_{j}b\|.$ (8.1)

Let the two Banach triples $B_{2}\subset B_{1}\subset B$ and $B_{2b}\subset
B_{1b}\subset B_{b}$ (with the norms of the second denoted $\|.\|_{2b}$,
$\|.\|_{1b}$ and $\|.\|_{b}$) be generated by the operators
$D=(D_{1},\cdots,D_{n})$ and $D^{b}=(D^{b}_{1},\cdots,D^{b}_{n})$
respectively. Let us say that the triples are dual if $(B,B_{b})$ forms a dual
pair and operators $D_{j}$ and $-D^{b}_{j}$ are dual, in the sense that

$(D_{j}b,f)=-(b,D^{b}_{j}f),\quad b\in B_{1},\,f\in B_{1b},\,j=1,\cdots,n.$

With some abuse of notation, we shall write just $D$ also for the operator
$D^{b}$, so that the previous equation becomes $(D_{j}b,f)=-(b,D_{j}f)$, and
we shall say that the dual triples are generated by $D$. Of course in all
applications, both $D$ and $D^{b}$ are the gradients, so that this unified
notation is fully natural.

When discussing duality it is useful to work with two versions of the
smoothing property (4.5). Namely, let the Banach triple $B_{2}\subset
B_{1}\subset B$ be generated by $D=(D_{1},\cdots,D_{n})$ and $A$ generate a
strongly continuous semigroup $e^{At}$ in $B$. Let us say that the semigroup
$e^{At}$ is smoothing from the right in the 1st order, if (4.5) holds. Since
the triple is generated by $D$, this is equivalent to the estimate

$\|D_{j}e^{At}b\|\leq\varkappa t^{-\omega}\|b\|,\quad j=1,\cdots,n,\quad
t\in(0,1],$ (8.2)

for all $b\in B$ (possibly with another constant $\varkappa$ than in (4.5),
but with the same $\omega$). Let us say that the semigroup $e^{At}$ is
smoothing from the left in the 1st order, if the operators $e^{At}D_{j}$
defined on $B_{1}$ can be extended to bounded operators in $B$ for any $t>0$
such that

$\|e^{At}D_{j}b\|\leq\varkappa t^{-\omega}\|b\|,\quad j=1,\cdots,n,\quad
t\in(0,1],$ (8.3)

for all $b\in B$.

For a Banach triple generated by $D$ let us say that the semigroup $e^{At}$ is
smoothing from the right in the 2nd order, if

$\|D_{j}e^{At}b\|_{1}\leq\varkappa t^{-\omega}\|b\|_{1},\quad
j=1,\cdots,n,\quad t\in(0,1],$ (8.4)

holds, and it is smoothing from the left in the 2nd order, if

$\|e^{At}D_{j}b\|_{1}\leq\varkappa t^{-\omega}\|b\|_{1},\quad
j=1,\cdots,n,\quad t\in(0,1],$ (8.5)

holds.

The next result supplies some links between various smoothing properties.

######  Theorem 8.1.

Let the two Banach triples $B_{2}\subset B_{1}\subset B$ and $B_{2b}\subset
B_{1b}\subset B_{b}$, generated by the operators $D=(D_{1},\cdots,D_{n})$, be
dual. Let the operators

$A\in\mathcal{L}(B_{2},B),\quad A^{b}\in\mathcal{L}(B_{2b},B_{b})$

be dual, $A^{b}=A^{*}$, in the sense that

$(Ab,f)=(b,A^{*}f),\quad b\in B_{2},\,f\in B_{2b},$

and let $A$ and $A^{b}=A^{*}$ generate strongly continuous semigroups $e^{At}$
in $B$ and $e^{A^{*}t}$ in $B_{b}$, having cores contained in $B_{1}$ and
$B_{1b}$ respectively.

(i) If $e^{At}$ is smoothing in the 1st order from the right (respectively,
from the left), then $e^{A^{*}t}$ is smoothing in the 1st order from the left
(respectively, from the right), with the same parameter $\omega$, and vice
versa.

(ii) If the commutators $[D_{j},e^{At}]$ extend to uniformly bounded (in
$t\in[0,T]$ for any $T$) operators in $B$, then (a) $e^{At}$ is a bounded
semigroup in $B_{1}$ (that is, the second estimate of (4.1) follows from the
1st one) and (b) $e^{At}$ is smoothing in the 1st order from the right if and
only if it is smoothing in the 1st order from the left.

(iii) $[D_{j},e^{At}]$ extend to uniformly bounded operators in $B$ if and
only if $[D_{j},e^{A^{*}t}]$ extend to uniformly bounded operators in $B_{b}$.

(iv) If $e^{A^{*}t}$ is smoothing from the left of the 1st order, then
$e^{At}$ is smoothing from the left in the 2nd order.

(v) If the commutators $[D_{j},e^{At}]$ extend to uniformly bounded (in
$t\in[0,T]$ for any $T$) operators in $B_{1}$, then (a) $e^{At}$ is a bounded
semigroup in $B_{2}$ and (b) $e^{At}$ is smoothing in the 2nd order from the
right if and only if it is smoothing in the 2nd order from the left.

###### Proof.

(i) Let $e^{At}$ be smoothing in the 1st order from the right. If $f\in
B_{1b}$ belongs to the core of $e^{A^{*}t}$, then

$\|D_{j}^{b}e^{A^{*}t}f\|_{b}=\sup_{\|b\|\leq
1}|(b,D_{j}^{b}e^{A^{*}t}f)|=\sup_{\|b\|\leq 1,b\in
B_{1}}|(b,D_{j}^{b}e^{A^{*}t}f)|=\sup_{\|b\|\leq 1,b\in
B_{1}}|(D_{j}b,e^{A^{*}t}f)$ $=\sup_{\|b\|\leq 1,b\in
B_{1}}|(e^{At}D_{j}b,f)|\leq\varkappa e^{-\omega t}\|f\|_{b}.$

Since the core is dense, the operator $D_{j}e^{A^{*}t}$ extends to the bounded
operator $B_{b}\to B_{b}$. Similarly, other statements are proved.

(ii) (a) If $b\in B_{1}$, then for $t\in[0,T]$,

$\|D_{j}e^{At}b\|\leq\|e^{At}D_{j}b\|+\|[D_{j},e^{At}]b\|\leq
Me^{mt}\|b\|_{1}+C(T)\|b\|.$

and thus

$\|e^{At}b\|_{1}\leq
nMe^{mt}\|b\|_{1}+(1+C(T))\|b\|\leq(nMe^{mt}+1+C(T))\|b\|_{1},$

so that the operators $e^{At}$ are bounded in $B_{1}$ uniformly for
$t\in[0,T]$. Statement (b) is straightforward.

(iii) This follows from the duality relation:

$([D_{j},e^{At}]b,f)=(b,[D_{j},e^{A^{*}t}]f).$

(iv) To estimate $\|e^{At}D_{j}b\|_{1}$ we need to estimate
$\|D_{i}e^{At}D_{j}b\|$. We have

$\|D_{i}e^{At}D_{j}b\|=\sup_{f\in B_{1b},\|f\|_{b}\leq
1}|(D_{i}e^{At}D_{j}b,f)|=\sup_{f\in B_{1b},\|f\|_{b}\leq
1}|(D_{j}b,e^{A^{*}t}D_{i}f)|$ $\leq\|b\|_{1}\sup_{\|f\|_{b}\leq
1}\|e^{A^{*}t}D_{i}f\|_{b}\leq\varkappa t^{-\omega}\|b\|_{1}.$

(v) (a) In order to estimate $\|e^{At}b\|_{2}$, we need the estimate for
$\|D_{i}D_{j}e^{At}b\|$. We have

$\|D_{i}D_{j}e^{At}b\|=\|D_{i}e^{At}D_{j}b\|+\|D_{i}[e^{At},D_{j}]b\|\leq\|e^{At}D_{i}D_{j}b\|+\|[D_{i},e^{At}]D_{j}b\|+\|D_{i}[e^{At},D_{j}]b\|$
$\leq\|e^{At}\|_{B\to B}\|b\|_{2}+\|[D_{i},e^{At}]\|_{B_{1}\to
B_{1}}\|b\|_{2}+\|[D_{i},e^{At}]\|_{B_{1}\to B_{1}}\|b\|_{1}$
$\leq\left(\|e^{At}\|_{B\to B}+2\|[D_{i},e^{At}]\|_{B_{1}\to
B_{1}}\right)\|b\|_{2}.$

(v) (b) Assume that $e^{At}$ is smoothing in the 2nd order from the left. To
show that it is smoothing in the 2nd order from the right, we need to estimate
$\|D_{j}e^{At}b\|_{1}$ and thus $\|D_{i}D_{j}e^{At}b\|$. We have

$\|D_{i}D_{j}e^{At}b\|\leq\|D_{i}e^{At}D_{j}b\|+\|D_{i}[D_{j},e^{At}]b\|,$

and both terms are estimated by $\|b\|_{1}$, as required. ∎

For dual Banach triples and dual generators, we can reformulate Theorems 6.1
and 7.1 almost without any additional assumptions on $A^{*}$. The following
result is a direct consequence of Theorems 6.1, 7.1 and 8.1.

######  Theorem 8.2.

Let the two Banach triples $B_{2}\subset B_{1}\subset B$ and $B_{2b}\subset
B_{1b}\subset B_{b}$, generated by the operators $D=(D_{1},\cdots,D_{n})$, be
dual. Let the operators

$A\in\mathcal{L}(B_{2},B),\quad A^{b}\in\mathcal{L}(B_{2b},B_{b})$

be dual, $A^{b}=A^{*}$, in the sense that

$(Ab,f)=(b,A^{*}f),\quad b\in B_{2},\,f\in B_{2b},$

and let $A$ and $A^{b}=A^{*}$ generate strongly continuous semigroups $e^{At}$
in $B$ and $e^{A^{*}t}$ in $B_{b}$, having cores contained in $B_{1}$ and
$B_{1b}$ respectively. Moreover, $A$ is smoothing of the 1st order from the
right or from the left and the operators $[e^{At},D_{j}]$ extend to uniformly
bounded operators $B\to B$. Let the assumptions (iii) and (iv) about
$H,H^{b}$, $u$ of conditions (AFB) hold. Then, for any $Y\in B_{1}$, $Z\in
B_{1b}$ there exists $T_{0}$ such that the forward-backward systems (6.6) and
(7.1) with $A^{b}=A^{*}$ have unique mild solutions for all $T<T_{0}$.

Let us specify the abstract setting to a more concrete forward-backward system
consisting of the coupled McKean Vlasov (forward) and HJB (backward) equations
for $t\in[a,T]$:

$\left\\{\begin{aligned}
&D_{a+*}g(t,x)=A^{*}g(t,x)+\sum_{j=1}^{d}h_{j}(t,g(t,.),u(t,x,\frac{\partial
f}{\partial x}(t,x)))\frac{\partial g}{\partial x_{j}},\quad g(a)=Y,\\\
&D_{T-*}f(t,x)=-Af(t,x)+H(t,x,\frac{\partial f}{\partial x}(t,x),g_{\geq
t}),\quad f(T)=Z.\end{aligned}\right.$ (8.6)

Here the dual Banach triples are $W_{2}(\mathbf{R}^{d})\subset
W_{1}(\mathbf{R}^{d})\subset L_{1}(\mathbf{R}^{d})$ for the first (forward)
equation and $C^{2}_{\infty}(\mathbf{R}^{d})\subset
C^{1}_{\infty}(\mathbf{R}^{d})\subset C_{\infty}(\mathbf{R}^{d})$ for the
second (backward) equation, both generated by the derivative operator
$D=\partial/\partial x$. The corresponding norms were defined at the end of
Section 4. The Banach space $B^{con}$ from assumption (AFB) is the space of
$U$-valued continuous functions on $R^{d}$, $U$ a compact subset of Euclidean
space. At the end of Section 4 were given examples of the generators $A$ that
satisfy the requirements of Theorem 8.2, which supplies the conditions for
local well-posedness for system (8.6).

##  9 Fractional McKean-Vlasov and HJB equations on manifolds

Let

$\Delta_{LB}\phi={div}\,(\nabla\phi)=\frac{1}{\sqrt{\det
g}}\sum_{j,k}\frac{\partial}{\partial x_{j}}\left(\sqrt{\det
g}\,g^{jk}\frac{\partial}{\partial x_{k}}\right)$ (9.1)

denote the Laplace-Beltrami operator on a compact Riemannian manifold $(M,g)$
of dimension $N$, with the Riemannian metric given by the matrix
$g=(g_{jk}(x))$ and its inverse matrix $G=(g^{jk}(x))$. Let $K(t,x,y)$ be the
corresponding heat kernel, that is, $K(t,x,y)$ is the solution of the heat
equation $(\partial K/\partial t)=\Delta_{LB}K$ as a function of $(t>0,x\in
M)$ and has the Dirac initial condition $K(0,x,y)=\delta_{y}(x)$. It is well
known that the Cauchy problem for this heat equation is well posed in $M$ and
the resolving operators

$S_{t}f(x)=e^{t\Delta_{LB}}=\int_{M}K(t,x,y)f(y)\,dv(y),$ (9.2)

where $dv(y)$ is the Remannian volume on $M$, form a strongly continuous
semigroup of contractions (the Markovian semigroup of the Brownian motion in
$M$) in the space $C(M)$ of bounded continuous functions on $M$, equipped with
the sup-norm.

Let $C^{1}(M)$ denote the space of continuously differentiable functions on
$M$ equipped with the norm

$\|f\|_{C^{1}(M)}=\|f\|_{C(M)}+\sup_{x}\|\nabla f(x)\|_{x},$ (9.3)

where in local coordinates

$\|\nabla f(x)\|_{x}^{2}=\left(\nabla f(x),G(x)\nabla
f(x)\right)=\sum_{jk}g^{jk}(x)\frac{\partial f}{\partial x_{j}}\frac{\partial
f}{\partial x_{k}},$ (9.4)

and $C^{2}(M)$ denote the space of twice continuously differentiable functions
on $M$ equipped with the norm

$\|f\|_{C^{2}(M)}=\|f\|_{C^{1}(M)}+\sup_{x}\|\nabla^{2}f(x)\|_{x},$ (9.5)

where in local coordinates

$\|\nabla^{2}f(x)\|_{x}^{2}=\sum_{jk}g^{jk}(x)g^{im}(x)\frac{\partial^{2}f}{\partial
x_{j}\partial x_{i}}\frac{\partial^{2}f}{\partial x_{k}\partial x_{m}}.$ (9.6)

The gradient $\nabla f(x)$ is an element of $T^{*}M_{x}$, the cotangent space
to $M$ at $x$, and $\nabla^{2}f$ is a tensor of type $(0,2)$. Formulas (9.4)
and (9.6) represent the standard lifting of the Riemannian metric to tensors.

The dual Banach triples used for the analysis of equations on manifolds are
the natural analogs of the triples used for equations in $\mathbf{R}^{d}$.
Namely, these are the triple $C_{2}(M)\subset C_{1}(M)\subset C(M)$ (with the
norms introduced above) and the triple of Sobolev functional spaces
$W_{2}(M)\subset W_{1}(M)\subset L_{1}(M)$, with the norms

$\|f\|_{L_{1}(M)}=\int_{M}|f(x)|\,dv(x),\quad\|f\|_{W_{1}(M)}=\|f\|_{L_{1}(M)}+\int_{M}\|\nabla
f(x)\|_{x}\,dv(x),$
$\|f\|_{W_{2}(M)}=\|f\|_{W_{1}(M)}+\int_{M}\|\nabla^{2}f(x)\|_{x}\,dv(x).$

It is known that the semigroup $S(t)$ in $C(M)$ has $C^{2}(M)$ as its
invariant core. It is also strongly continuous as a semigroup in $L_{1}(M)$
(actually in all $L_{p}(M)$, $p\geq 1$), see [2] and references therein.

######  Remark 9.1.

The identification of $\nabla f$ with the collection of $n$ functions
$\partial f/\partial x_{j}$, which is required to comply with the setting of
Sections 4 and 8, can be justified only in local coordinates, but not
globally. In order to have invariant theory we cannot assume that the operator
$D=(D_{1},\cdots,D_{n})$ act from $B=C(M)$ to $B^{n}$. In the invariant
exposition the gradient $D=\nabla$ maps $C(M)$ to the space $FT^{*}M$ of
covector fields on $M$, and $D^{2}=\nabla^{2}$ maps $C(M)$ to the symmetric
tensor fields of type $(0,2)$. However, the topologies of $C^{1}(M)$ and
$C^{2}(M)$ (as well as $W_{1}(M)$ and $W_{2}(M)$) remain to be generated by
$D=\nabla$ and $D^{2}$ respectively, according to (9.3), (9.5), in analogy
with (8.1). Thus the whole theory is recovered in this slightly generalized
setting.

The key smoothing and smoothness preservation properties of this semigroup
needed for our theory are collected in the next result.

######  Theorem 9.1.

(i) The operators $S_{t}$ are smoothing from the right and from the left:

$\|\nabla S_{t}f\|_{C(M)}\leq Ct^{-1/2}\|f\|_{C(M)},\quad\|S_{t}\nabla
f\|_{C(M)}\leq Ct^{-1/2}\|f\|_{C(M)},$ (9.7)

with a constant $C$, uniformly for any compact interval of time.

(ii) The operators $S_{t}$ are smoothness preserving:

$\|S_{t}f\|_{C^{1}(M)}\leq C\|f\|_{C^{1}(M)},$ (9.8)

with a constant $C$, uniformly for any compact interval of time.

(iii) The commutators $[\nabla,S_{t}]$ extend to bounded operators in $C(M)$,
uniformly for any compact interval of time.

###### Proof.

The first estimate in (9.7) is a consequence of the well-known estimate for
the derivatives of the heat kernel on a compact Riemannian manifold (see
Theorem 6 in [5]):

$\|\nabla K(t,x,y)\|_{M}\leq
C(\delta,N)t^{-N/2}t^{-1/2}\exp\left\\{-\frac{d^{2}(x,y)}{(4+\delta)t}\right\\},$
(9.9)

with any $\delta>0$ and a constant $C(\delta,N)$, where the derivative
$\nabla$ is taken with respect to $x$, and where $d$ is the Riemannian
distance in $M$. In fact, differentiating (9.2) and using (9.9) yields the
first estimate (9.7).

As shown in Theorem 8.1, the second estimate in (9.7) follows from the first
estimate and (iii). Also (ii) follows from (iii). Thus it remains to show
(iii). And this follows from (9.9) and the method of parametric (frozen
coefficients) approximation. This method (see e.g. formula (5.60) in [13])
starts by representing $K$ in terms of its asymptotics $K_{as}$ and the
integral correction as

$K(t,x,y)=K_{as}(t,x,y)+\int_{0}^{t}K(t-s,x,z)F(s,z,y)\,ds,$ (9.10)

where $F$ is the error term in the equation for $K_{as}$, that is

$\frac{\partial K_{as}}{\partial
t}(t,x,y)-\Delta_{LB}K_{as}(t,x,y)=-F(t,x,y).$

From (9.9) it follows that the derivative of the second term in (9.10) is
bounded and thus the statement about commutator reduces to the statement that
the integral operator with the integral kernel

$\frac{\partial K_{as}}{\partial x}(t,x,y)-\frac{\partial K_{as}}{\partial
y}(t,x,y)$

is bounded in $C(M)$. And this is performed as in the case of heat equations
in $\mathbf{R}^{d}$, as it is implied by the fact that $K_{as}$ is Gaussian
with ’frozen coefficients’. ∎

The following result is the direct consequence of Theorems 9.1 and 8.1, and
the well known fact that the operator $\Delta_{LB}$ is self-dual with respect
to the coupling given by the integration with respect to the volume measure
$dv$ on $M$.

######  Theorem 9.2.

The smoothing properties of the operators $S(t)$ also hold in the integral
norms, that is,

$\|\nabla S_{t}f\|_{L_{1}(M)}\leq Ct^{-1/2}\|f\|_{L_{1}(M)},\quad\|S_{t}\nabla
f\|_{L_{1}(M)}\leq Ct^{-1/2}\|f\|_{L_{1}(M)},$ (9.11)
$\|S_{t}f\|_{W_{1}(M)}\leq C\|f\|_{W_{1}(M)},$ (9.12)

and the commutators $[\nabla,S_{t}]$ extend to bounded operators in
$L_{1}(M)$, uniformly for any compact interval of time.

For the stochastic control of diffusions on $(M,g)$ with the second order part
being fixed as $\Delta_{LB}$, and where control is carried out via the drift
only (drift control of the Brownian motion on $M$), the corresponding HJB
equation is the equation

$\frac{\partial f}{\partial t}(t,x)=\Delta_{LB}f(t,x)+H(x,\nabla f(t,x)),$
(9.13)

where the Hamiltonian $H(x,p)$, $x\in M,p\in T^{*}M_{x}$ is a function on the
cotangent bundle $T^{*}M$ of the form

$H(x,p)=\sup_{u\in U}[(g(x,u),p)+J(x,u)],$ (9.14)

where $U$ is a compact set of possible controls and $J,g$ are some continuous
functions and a vector field depending on $u$ as a parameter. In case of zero-
sum stochastic two-player games with the so-called Isaac’s condition, the
Hamiltonian function takes the form

$H(x,p)=\sup_{u\in U}\inf_{v\in V}[(g(x,u,v),p)+J(x,u,v)]=\inf_{v\in
V}\sup_{u\in U}[(g(x,u,v),p)+J(x,u,v)].$ (9.15)

The possibility to exchange $\sup$ and $\inf$ here is called Isaac’s
condition. It is fulfilled, in particular, when the control of two players can
be separated in the sense that the Hamiltonian becomes

$H(x,p)=\sup_{u\in U}[(g_{1}(x,u),p)+J_{1}(x,u)]+\inf_{v\in
V}[(g_{2}(x,v),p)+J_{2}(x,v)]+J_{0}(x).$ (9.16)

The fractional version of equation (9.13) is the fractional HJB equation

$D_{a+*}f(t,x)=\Delta_{LB}f(t,x)+H(x,\nabla f(t,x))$ (9.17)

(fractional derivative acts on the variable $t$ and $\nabla$ and $\Delta_{LB}$
act on the variable $x\in M$), which describes the cost functions for the
scaled continuous-time random walks with the spatial motion govern by
$\Delta_{LB}$ (see [18]).

The McKean-Vlasov equation on $M$ describing a nonlinear diffusion on $M$
(with the operator $\Delta_{LB}$ as the main part and a nonlinear drift $h$)
is the equation of the form

$\frac{\partial f}{\partial t}(t,x)=\Delta_{LB}f(t,x)+(h(x,f(t,.)),\nabla
f(t,x)),$ (9.18)

where $h$ maps pairs $(x,f(t,.))$ to the elements of the tangent space
$T_{x}M$ of $M$ at $x$ (so that the coupling $(h(x,f(t,.)),\nabla f(t,x))$
yields a well defined function on $M$). Thus $h$ can be looked at as a vector
field on $M$ depending on $f$ as a parameter. The dependence of $h$ on $f$ via
certain integrals, that is,

$h(x,f(t,.))=h(x,\int_{M}g(x)f(t,x)\,dv(y)),$ (9.19)

where $g$ is some bounded function on $M$, is typical in applications.

The fractional version of equation (9.18) is, of course, the equation

$D_{a+*}f(t,x)=\Delta_{LB}f(t,x)+(h(x,f(t,.)),\nabla f(t,x)).$ (9.20)

The following results are direct consequences of Theorems 9.1, 9.2, 4.1 and
4.2 (and Remark 9.1).

######  Theorem 9.3.

(i) Let $H(x,p)$ be a continuous function on the cotangent bundle $T^{*}M$ to
the compact Riemannian manifold $(M,g)$ such that

$|H(x,p_{1})-H(x,p_{2})|\leq L_{H}\|p_{1}-p_{2}\|_{x}$ (9.21)

for all $x$ with a constant $L_{H}$. Then for any $Y\in C^{1}(M)$ there exists
a unique mild solution for the Cauchy problem of equation (9.13) and a unique
solution for the Cauchy problem of equation (9.17) with initial condition
$f(0,.)=Y$. These solutions depend Lipschitz continuously on the initial data
in the norm of $C^{1}(M)$.

(ii) Let $h(x,f(.))$ be a uniformly bounded vector field on $M$ depending
Lipschitz continuously on $f$ in the norm of $L_{1}(M)$:

$\|h(x,f_{1}(t,.))-h(x,f_{2}(t,.))\|_{x}\leq
L_{h}\|f_{1}(t,.)-f_{2}(t,.)\|_{L_{1}(M)},$ (9.22)

for all $x\in M$ with a constant $L_{h}$. Then for any $Y\in W_{1}(M)$ there
exists a unique mild solution for the Cauchy problem of equation (9.18) and a
unique solution for the Cauchy problem of equation (9.20) with initial
condition $f(0,.)=Y$. These solutions depend Lipschitz continuously on the
initial data in the norm of $W_{1}(M)$.

##  10 Fractional forward-backward systems on manifolds

Finally, we analyse the analogue of system (8.6) on manifolds with the
generator $A=\Delta_{LB}$:

$\left\\{\begin{aligned}
&D_{a+*}g(t,x)=\Delta_{LB}g(t,x)+(h(t,x,g(t,.),u(t,x,\nabla f(t,x)))\nabla
g(t,x),\quad g(a)=Y,\\\ &D_{T-*}f(t,x)=-\Delta_{LB}f(t,x)+H(t,x,\nabla
f(t,x),g_{\geq t}),\quad f(T)=Z.\end{aligned}\right.$ (10.1)

The following result is a consequence of Theorems 8.2, 9.1 and 9.2.

######  Theorem 10.1.

Let $U$ be a compact set in a Euclidean space and $u(t,x,p)\in U$ be a
continuous function of the triple $t\in\mathbf{R}$, $x\in M$, $p\in
T^{*}M_{x}$ which is Lipschitz continuous in the last argument:

$|u(t,x,p)-u(t,x,\tilde{p})|\leq L_{u}\|p-\tilde{p}\|_{x}.$ (10.2)

Let $h:\mathbf{R}\times M\times L_{1}(M)\times U\to TM$ be a continuous
mapping such that $h(t,x,f,u)\in TM_{x}$ for all $x,f,u$, which is Lipschitz
in the sense that

$\|h(t,x,g(t,.),u)-h(t,x,\tilde{g}(t,.),\tilde{u})\|_{x}\leq
L_{h}\|g(t,.)-\tilde{g}(t,.)\|_{L_{1}(M)}+L_{hu}|u-\tilde{u}|,$ (10.3)

Let $H(t,x,p,g_{\geq t})$, $t\in\mathbf{R},x\in M,p\in T^{*}M_{x},g\in
L_{1}(M)$ be a continuous function, which is Lipschitz in the sense that

$|H(t,x,p,g(.))-H(t,x,\tilde{p},g(.))|\leq L_{H}\|p-\tilde{p}\|_{x},$ (10.4)
$|H(t,x,p,g(.))-H(t,x,p,\tilde{g}(.))|\leq
L_{H}\|f-\tilde{f}\|_{C([a,T],B)}(1+\sum_{j=0}^{n}\|f_{j}\|_{b}).$ (10.5)

Then for all $Y,Z$ there exist $T_{0}$ such that the system (10.1) has a
unique solution for all $T\leq T_{0}$.

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# Contact Optimization with Learning from Demonstration:
Application in Long-term Non-prehensile Planar Manipulation

Teng Xue and Sylvain Calinon The authors are with the Robot Learning and
Interaction Group, Idiap Research Institute, Martigny, Switzerland and EPFL,
Lausanne, Switzerland; (e-mail<EMAIL_ADDRESS>
###### Abstract

Long-term non-prehensile planar manipulation is a challenging task for
planning and control, requiring determination of both continuous and discrete
contact configurations, such as contact points and modes. This leads to the
non-convexity and hybridness of contact optimization. To overcome these
difficulties, we propose a novel approach that incorporates human
demonstrations into trajectory optimization. We show that our approach
effectively handles the hybrid combinatorial nature of the problem, mitigates
the issues with local minima present in current state-of-the-art solvers, and
requires only a small number of demonstrations while delivering robust
generalization performance. We validate our results in simulation and
demonstrate its applicability on a pusher-slider system with a real Franka
Emika robot.

##  I Introduction

Motion planning for contact-rich manipulation is a complex task, especially in
the presence of non-convexity and hybridness. This requires avoiding poor
local optima and optimization over both continuous and discrete variables. We
refer to this as contact optimization. In this work, we focus on long-term
non-prehensile planar manipulation, which concerns joint logistic and
geometric planning and feedback control over diverse interaction modes and
face switches. For example, to push an object, a prerequisite is to decide how
much force should be applied, and which point to push. Moreover, in some cases
such as pushing an object with small distance but large orientation, relying
on a single fixed face is not feasible. Therefore, a sequence of face
switching is required, as well as contact mode schedule resulting from Coulomb
friction.

To solve non-convex problems, trajectory optimization methods are widely used.
They iteratively optimize around the current solution till reaching local
optima. Although these approaches can yield good practical results, the
convergence and solution quality are quite sensitive to the initial guess. The
methods can easily get stuck in poor local optima if the initial guess is far
from the optimal solution. Moreover, these methods struggle with the hybrid
nature of contact. Gradients do not exist when making and breaking contacts,
making powerful continuous solvers unusable. One common strategy is to model
the contact with complementary constraints on continuous variables [1].
However, these constraints can render the optimization problem nearly ill-
conditioned, and still susceptible to poor local optima. To address this
issue, researchers have proposed warm-starting trajectory optimization methods
that provide good initializations to solvers [2]. In this paper, we introduce
human demonstration as trajectory constraints to the optimization framework,
combining the benefits of both approaches. Our approach effectively handles
the hybrid variables that arise in contact-rich manipulation tasks while
avoiding poor local optima.

Figure 1: Left: Multiple contact modes, including sticking (ST), Sliding Up
(SU), and Sliding Down (SD) modes for each contact face, as well as a
separation (SE) mode for face switching. Middle: Demonstration interface. The
dotted square represents the target and the red triangle indicates the real-
time position of the mouse. Humans can manipulate the mouse to guide the
pusher during the demonstration. Right: System setup where the robot pushes an
object with changes of contact points and faces.

##  II Dynamical System

We first build the dynamics model for the pusher-slider system with face-
switching strategy, namely

$\bm{\dot{x}}=\left\\{\begin{aligned}
&\bm{g}_{1}(\bm{x},\bm{u}),\quad\text{if}\quad\textbf{Sticking},\\\
&\bm{g}_{2}(\bm{x},\bm{u}),\quad\text{if}\quad\textbf{Sliding Up},\\\
&\bm{g}_{3}(\bm{x},\bm{u}),\quad\text{if}\quad\textbf{Sliding Down},\\\
&\bm{g}_{4}(\bm{x},\bm{u}),\quad\text{if}\quad\textbf{Separation},\end{aligned}\right.$
(1)

where $\bm{x}=[\bm{q}_{s}^{\scriptscriptstyle\top}\
\bm{q}_{p_{s}}^{\scriptscriptstyle\top}\
\bm{v}^{\scriptscriptstyle\top}]^{\scriptscriptstyle\top}$, representing the
slider pose, pusher position and velocity. $\bm{u}$ denotes the acceleration
of the pusher. For more information, please refer to [3].

The system, as illustrated in Fig. 1-Left, consists of three different contact
modes (sticking, sliding up and down) for each contact face, along with one
separation mode for face switching. Consequently, the problem involves 13
interaction modes and several continuous variables in both state and action
spaces.

##  III Methods and Results

###  III-A Problem formulation

An optimal control problem (OCP) can be described as

$\displaystyle\min_{\bm{u}_{t}}\quad
c_{T}(\bm{x}_{T})+\sum_{t=0}^{T-1}c_{t}(\bm{x}_{t},\bm{u}_{t}),$ (2)
$\displaystyle\begin{array}[]{r@{\quad}r@{}l@{\quad}l}\text{s.t.}\quad\bm{x}_{t+1}=\bm{f}(\bm{x}_{t},\bm{u}_{t}),\end{array}$
(4)

where (2) is the cost function and (4) is the dynamic equation. We employ DDP
[4] to solve this problem, as it can provide a local feedback mechanism for
the controller.

###  III-B Demonstration-started DDP (DS-DDP)

Since (2) is a non-convex problem, DDP optimizes around the current solution
iteratively to solve it. However, as Table I shows, convergence is highly
dependent on the initial guess, where ZS-DDP represents the typical DDP
without demonstration. To address this issue, we incorporate human
demonstration as an initialization into the optimization framework. Fig.
1-Middle shows the demonstration interface, which allows humans to guide the
motion of the pusher in real-time using a mouse. The motion of the slider is
simulated based on the dynamics model (1). For more details, please see our
webpage111https://sites.google.com/view/dg-oc/. We collected demonstration
trajectories for both the pusher and slider for three different targets,
corresponding to $N_{s}=0$, $N_{s}=1$, and $N_{s}=2$, respectively, where
$N_{s}$ is the number of face switches during the demonstration. Given a
target, we use K-nearest neighbor (k-NN) to select the closest demonstration
from the set.

The cost function of DS-DDP is defined as

$\begin{gathered}DS_{cost}=c_{re}+c_{rg}+c_{bd},\end{gathered}$ (5)

where
$c_{re}=(\bm{\mu}_{T}-\bm{x}_{T})^{\scriptscriptstyle\top}\bm{Q}_{T}(\bm{\mu}_{T}-\bm{x}_{T})$
represents the reaching cost,
$c_{rg}=\sum_{t=0}^{T-1}\bm{u}_{t}^{\scriptscriptstyle\top}\bm{R}\bm{u}_{t}$
is the regularizer for control commands, and
$c_{bd}=\sum_{t=0}^{T-1}\bm{f}^{\text{cut}}(\bm{u}_{t},\bm{u}_{l})^{\scriptscriptstyle\top}\bm{Q}_{f}\bm{f}^{\text{cut}}(\bm{u}_{t},\bm{u}_{l})$
penalizes control commands through a soft-thresholding function
$\bm{f}^{\text{cut}}$. Here, $\bm{\mu}_{T}$ is the target state, and
$\bm{u}_{l}$ is the boundary of $\bm{u}$. The initial guess
${\bm{u}^{0}}=\bm{\widetilde{u}}$ is drawn from human demonstrations directly.

###  III-C Demonstration-penalized DDP (DP-DDP)

The quality and quantity of demonstrations can significantly impact the
optimization outcome when using them as the sole initialization. However,
demonstrations can also serve as constraints or guidance for optimization,
akin to mathematical programming with complementary constraints. Instead of
adding slack variables to hard constraints for contact relaxation, we
incorporate soft constraints by penalizing the cost function. By leveraging
demonstrations as constraints, we can guide the trajectory optimization and
mitigate the possibility of getting trapped in poor local optima while
effectively handling hybrid variables. It is achieved by designing the cost
function as

$DP_{cost}=c_{re}+c_{rg}+c_{bd}+c_{sw}+c_{ve}+c_{ac},$ (6)

with
$c_{sw}=\sum_{n=t_{0}}^{t_{N-1}}(\bm{\mu}_{n}-\bm{x}_{n})^{\scriptscriptstyle\top}\bm{Q}_{n}(\bm{\mu}_{n}-\bm{x}_{n})$,
$c_{ve}=\sum_{t=0}^{T-1}(\bm{\widetilde{v}}_{t}-\bm{v}_{t})^{\scriptscriptstyle\top}\bm{R}_{dv}(\bm{\widetilde{v}}_{t}-\bm{v}_{t})$
and
$c_{ac}=\sum_{t=0}^{T-1}(\bm{\widetilde{u}}_{t}-\bm{u}_{t})^{\scriptscriptstyle\top}\bm{R}_{du}(\bm{\widetilde{u}}_{t}-\bm{u}_{t})$,
where $c_{sw}$, $c_{ve}$ and $c_{ac}$ are designed to follow the demonstrated
face-switching strategy, pusher velocity and pusher acceleration.
$n=[t_{0},\cdots,t_{N-1}]$ is the timestep when the contact face switches,
$\bm{\mu}$ is the state of the selected demonstration, and
$\bm{\widetilde{v}}_{t}$, $\bm{\widetilde{u}}_{t}$ are the demonstrated
velocity and acceleration at timestep $t$.

###  III-D Warm-starting DDP (WS-DDP)

Demonstration-penalized DDP shows good performance and can avoid poor local
optima, but in order to further improve its convergence properties, we propose
a hierarchical optimization framework, where the solution of DP-DDP is used to
initialize another DDP problem that we call Warm-starting DDP (WS-DDP).

The cost function of WS-DDP is as same as DS-DDP, allowing it to explore much
freely towards the final target. The initial guess
${\bm{u}^{0}}=\bm{u}^{*}_{DP}$ is the solution of DP-DDP.

###  III-E Results

Table I: Performance of ZS-DDP, DS-DDP, DP-DDP, and WS-DDP for offline programming Method |  $\bm{x_{\text{err}}}$ (cm) |  $\bm{y_{\text{err}}}$ (cm) |  $\bm{\theta_{\text{err}}}$ (rad) | succ_rate
---|---|---|---|---
ZS-DDP |  3.18 $\pm$ 8.30 |  4.78 $\pm$ 8.60 |  0.04 $\pm$ 0.39 | 40%
DS-DDP |  1.70 $\pm$ 9.07 |  1.62 $\pm$ 9.66 |  0.21 $\pm$ 1.82 | 58%
DP-DDP |  0.14 $\pm$ 1.54 |  0.57 $\pm$ 2.77 |  0.22 $\pm$ 0.12 | 77%
WS-DDP | 0.07 $\pm$ 1.31 | 0.19 $\pm$ 2.90 | 0.01 $\pm$ 0.08 | 85%

To evaluate the effectiveness of our proposed method, we conducted experiments
on 100 randomly selected targets from the task space
$\mathbb{T}=\\{[x,y,\theta]:x\in[\scalebox{0.75}[1.0]{$-$}25\text{cm},25\text{cm}],y\in[\scalebox{0.75}[1.0]{$-$}25\text{cm},25\text{cm}],\theta\in[\scalebox{0.75}[1.0]{$-$}\pi,\pi]\\}$.
A successful contact planning is achieved when the end-effector reaches the
target with an error less than
$\\{x_{\text{err}}<1\text{cm},\;y_{\text{err}}<1\text{cm},\;\theta_{\text{err}}<5^{\circ}\\}$.
The statistical results are presented in Table I.

The success rate of our baseline method ZS-DDP is only 40%, indicating that it
is susceptible to getting stuck in poor local minima. However, by initializing
the optimization with demonstrations, we observe a success rate improvement of
nearly 20%, highlighting the effectiveness of using human demonstrations to
avoid poor local minima.

To further enhance the convergence properties of our method, we propose a
hierarchical optimization framework consisting of DP-DDP and WS-DDP. It
achieves a significantly higher success rate of 85% by incorporating human
demonstrations as soft constraints and warm-starting a new DDP. By using
demonstration as soft constraints, rather than initialization, our approach is
less dependent on demonstrations while retaining the prior information. This
results in more freedom to optimize around the demonstration trajectory. Our
results also show the strong generalization ability of our approach, which
only requires a few demonstrations.

The use of human demonstrations in contact-rich manipulation tasks enables
traditional trajectory optimization methods to handle both non-convexity and
hybridness, opening up promising avenues for further research. For instance,
life-long learning from demonstration could be used to learn the constraints
from a vast range of daily tasks. These constraints could then be integrated
into the optimization framework, enabling the system to develop a dexterous
manipulation policy that can handle various tasks robustly.

For a more in-depth understanding of this work, we encourage readers to refer
to [3], which presents an extended version with more detailed information.

## References

  * [1] M. Posa, C. Cantu, and R. Tedrake, “A direct method for trajectory optimization of rigid bodies through contact,” _IJRR_ , vol. 33, no. 1, pp. 69–81, 2014.
  * [2] N. Mansard, A. DelPrete, M. Geisert, S. Tonneau, and O. Stasse, “Using a memory of motion to efficiently warm-start a nonlinear predictive controller,” in _Proc. IEEE Intl Conf. on Robotics and Automation (ICRA)_. IEEE, 2018, pp. 2986–2993.
  * [3] T. Xue, H. Girgin, T. S. Lembono, and S. Calinon, “Demonstration-guided optimal control for long-term non-prehensile planar manipulation,” _arXiv preprint arXiv:2212.12814_ , 2022.
  * [4] D. Mayne, “A second-order gradient method for determining optimal trajectories of non-linear discrete-time systems,” _International Journal of Control_ , vol. 3, no. 1, pp. 85–95, 1966.