On Some Discontinuous Control Problems Dan Goreac 1 Universit - PowerPoint PPT Presentation

An example Deterministic framework Main result Stochastic framework Idea of the proof Idea of the proof of (c) 1 Choose x ε 2 cl ( R ( T , t 0 ) x 0 ) s.t. h ( x ε ) < V ( t 0 , x 0 ) + ε . De…ne h ε : R N � ! R l.s.c. � h ( x ε ) , if x = x ε , h ε ( x ) = sup x h ( x ) , otherwise. Dan Goreac On Some Discontinuous Control Problems

An example Deterministic framework Main result Stochastic framework Idea of the proof Idea of the proof of (c) 1 Choose x ε 2 cl ( R ( T , t 0 ) x 0 ) s.t. h ( x ε ) < V ( t 0 , x 0 ) + ε . De…ne h ε : R N � ! R l.s.c. � h ( x ε ) , if x = x ε , h ε ( x ) = sup x h ( x ) , otherwise. Value function V ε ( t , x ) = inf f h ε ( y ) : y 2 cl ( R ( T , t ) x ) g , satis…es: V ε ( t 0 , x 0 ) = h ( x ε ) � V ( t 0 , x 0 ) + ε Dan Goreac On Some Discontinuous Control Problems

An example Deterministic framework Main result Stochastic framework Idea of the proof Idea of the proof of (c) 2 De…ne g : R N � ! R u.s.c. � V ( t 0 , x 0 ) , if x 2 cl ( R ( T , t 0 ) x 0 ) , g ( x ) = inf y 2 R N h ( y ) , otherwise. Dan Goreac On Some Discontinuous Control Problems

An example Deterministic framework Main result Stochastic framework Idea of the proof Idea of the proof of (c) 2 De…ne g : R N � ! R u.s.c. � V ( t 0 , x 0 ) , if x 2 cl ( R ( T , t 0 ) x 0 ) , g ( x ) = inf y 2 R N h ( y ) , otherwise. Consider V g V g ( t , x ) = inf f g ( y ) : y 2 cl ( R ( T , t ) x ) g , Dan Goreac On Some Discontinuous Control Problems

An example Deterministic framework Main result Stochastic framework Idea of the proof Idea of the proof of (c) 2 De…ne g : R N � ! R u.s.c. � V ( t 0 , x 0 ) , if x 2 cl ( R ( T , t 0 ) x 0 ) , g ( x ) = inf y 2 R N h ( y ) , otherwise. Consider V g V g ( t , x ) = inf f g ( y ) : y 2 cl ( R ( T , t ) x ) g , V g ( T , � ) � g ( � ) � h ( � ) and V g ( t 0 , x 0 ) = V ( t 0 , x 0 ) Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation ( Ω , F , P ) complete probability Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation ( Ω , F , P ) complete probability a …ltration F = ( F t ) t � 0 satisfying the usual assumptions Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation ( Ω , F , P ) complete probability a …ltration F = ( F t ) t � 0 satisfying the usual assumptions W be a standard, d -dimensional Brownian motion Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation ( Ω , F , P ) complete probability a …ltration F = ( F t ) t � 0 satisfying the usual assumptions W be a standard, d -dimensional Brownian motion admissible (strong) control u 2 U : U -valued, progressively measurable Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation ( Ω , F , P ) complete probability a …ltration F = ( F t ) t � 0 satisfying the usual assumptions W be a standard, d -dimensional Brownian motion admissible (strong) control u 2 U : U -valued, progressively measurable � dX t , x , u = b ( s , X t , x , u , u s ) ds + σ ( s , X t , x , u , u s ) dW s , t � s � T , s s s X t , x , u = x 2 R N , t ( 2 ) Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Main assumptions b : R � R N � U � ! R N , σ : R � R N � U � ! R N � d Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Main assumptions b : R � R N � U � ! R N , σ : R � R N � U � ! R N � d (i) b , σ bounded, uniformly continuous Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Main assumptions b : R � R N � U � ! R N , σ : R � R N � U � ! R N � d (i) b , σ bounded, uniformly continuous (ii) 9 c > 0 s.t. j b ( t , x , u ) � b ( t , y , u ) j + j σ ( t , x , u ) � σ ( t , y , u ) j � c j x � y j and j b ( t , x , u ) � b ( s , x , u ) j + j σ ( t , x , u ) � σ ( s , x , u ) j � δ 0 2 , c j t � s j 8 ( t , s , x , y , u ) 2 [ 0 , T ] 2 � R 2 N � U . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Linear formulation in Lipschitz case Assume : h : R N � ! R is bounded and Lipschitz-continuous. Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Linear formulation in Lipschitz case Assume : h : R N � ! R is bounded and Lipschitz-continuous. � � �� X t , x , u Value function V h ( t , x ) = inf u 2U E h . T Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Linear formulation in Lipschitz case Assume : h : R N � ! R is bounded and Lipschitz-continuous. � � �� X t , x , u Value function V h ( t , x ) = inf u 2U E h . T V h is he unique viscosity solution in the class of linear-growth continuous functions of Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Linear formulation in Lipschitz case Assume : h : R N � ! R is bounded and Lipschitz-continuous. � � �� X t , x , u Value function V h ( t , x ) = inf u 2U E h . T V h is he unique viscosity solution in the class of linear-growth continuous functions of 8 � � = 0 , x , DV h ( t , x ) , D 2 V h ( t , x ) � ∂ t V h ( t , x ) + H < for all ( t , x ) 2 ( 0 , T ) � R N , (HJB) , : V h ( T , � ) = h ( � ) on R N , H ( t , x , p , A ) = � � 1 � 2 Tr ( σσ � ( t , x , u ) A ) � h b ( t , x , u ) , p i sup u 2 U , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 1 Linear programming tools: Stockbridge (90), Bhatt, Borkar (’96), Kurtz, Stockbridge (’98), Borkar, Gaitsgory (’05) Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 1 Linear programming tools: Stockbridge (90), Bhatt, Borkar (’96), Kurtz, Stockbridge (’98), Borkar, Gaitsgory (’05) ( t , x ) 2 [ 0 , T ) � R N , u 2 U , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 1 Linear programming tools: Stockbridge (90), Bhatt, Borkar (’96), Kurtz, Stockbridge (’98), Borkar, Gaitsgory (’05) ( t , x ) 2 [ 0 , T ) � R N , u 2 U , γ t , x , u ( A � B � C � D ) = h R T i � � 1 t 1 A � B � C (( s , X t , x , u X t , x , u T � t E , u s )) ds P 2 D , s T Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 1 Linear programming tools: Stockbridge (90), Bhatt, Borkar (’96), Kurtz, Stockbridge (’98), Borkar, Gaitsgory (’05) ( t , x ) 2 [ 0 , T ) � R N , u 2 U , γ t , x , u ( A � B � C � D ) = h R T i � � 1 t 1 A � B � C (( s , X t , x , u X t , x , u T � t E , u s )) ds P 2 D , s T A � B � C � D � [ 0 , T ] � R N � U � R N , Borel sets. Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 1 Linear programming tools: Stockbridge (90), Bhatt, Borkar (’96), Kurtz, Stockbridge (’98), Borkar, Gaitsgory (’05) ( t , x ) 2 [ 0 , T ) � R N , u 2 U , γ t , x , u ( A � B � C � D ) = h R T i � � 1 t 1 A � B � C (( s , X t , x , u X t , x , u T � t E , u s )) ds P 2 D , s T A � B � C � D � [ 0 , T ] � R N � U � R N , Borel sets. � j y j 2 + j z j 2 � R γ t , x , u ( ds , dy , dv , dz ) � [ t , T ] � R N � U � R N � � j x j 2 + 1 C 0 . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 1 Linear programming tools: Stockbridge (90), Bhatt, Borkar (’96), Kurtz, Stockbridge (’98), Borkar, Gaitsgory (’05) ( t , x ) 2 [ 0 , T ) � R N , u 2 U , γ t , x , u ( A � B � C � D ) = h R T i � � 1 t 1 A � B � C (( s , X t , x , u X t , x , u T � t E , u s )) ds P 2 D , s T A � B � C � D � [ 0 , T ] � R N � U � R N , Borel sets. � j y j 2 + j z j 2 � R γ t , x , u ( ds , dy , dv , dz ) � [ t , T ] � R N � U � R N � � j x j 2 + 1 C 0 . � [ t , T ] � R N � U � R N � γ t , x , u 2 P : 8 φ 2 � [ 0 , T ] � R N � C 1 , 2 , � � b ( T � t ) L v φ ( s , y ) R γ ( ds , dy , dv , dz ) = [ t , T ] � R N � U � R N + φ ( t , x ) � φ ( T , z ) 0 (Itô’s formula). Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 2 � � �� = X t , x , u J h ( t , x , u ) = E h R � [ t , T ] � R N � U , dz � T R N h ( z ) γ t , x , u . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 2 � � �� = X t , x , u J h ( t , x , u ) = E h R � [ t , T ] � R N � U , dz � T R N h ( z ) γ t , x , u . Θ ( t , x ) = 8 � [ t , T ] � R N � U � R N � � [ 0 , T ] � R N � : 8 φ 2 C 1 , 2 > γ 2 P , > � � b > ( T � t ) L v φ ( s , y ) < R γ ( ds , dy , dv , dz ) = 0, [ t , T ] � R N � U � R N + φ ( t , x ) � φ ( T , z ) > � j y j 2 + j z j 2 � � � > R > j x j 2 + 1 : γ ( ds , dy , dv , dz ) � C 0 [ t , T ] � R N � U � R N Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation (Finite horizon) Occupational measures 2 � � �� = X t , x , u J h ( t , x , u ) = E h R � [ t , T ] � R N � U , dz � T R N h ( z ) γ t , x , u . Θ ( t , x ) = 8 � [ t , T ] � R N � U � R N � � [ 0 , T ] � R N � : 8 φ 2 C 1 , 2 > γ 2 P , > � � b > ( T � t ) L v φ ( s , y ) < R γ ( ds , dy , dv , dz ) = 0, [ t , T ] � R N � U � R N + φ ( t , x ) � φ ( T , z ) > � j y j 2 + j z j 2 � � � > R > j x j 2 + 1 : γ ( ds , dy , dv , dz ) � C 0 [ t , T ] � R N � U � R N � ( σσ � ) ( s , y , v ) D 2 φ ( s , y ) � L v φ ( s , y ) = 1 2 Tr + h b ( s , y , v ) , D φ ( s , y ) i + ∂ t φ ( s , y ) , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Linearized formulation R � [ t , T ] , R N , U , dz � h � ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Linearized formulation R � [ t , T ] , R N , U , dz � h � ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , dual formulation: η � ( t , x ) = 8 9 � [ 0 , T ] � R N � η 2 R : 9 φ 2 C 1 , 2 < s.t. = b 8 ( s , y , v , z ) 2 [ t , T ] � R N � V � R N , sup ; , : η � ( T � t ) L v φ ( s , y ) + h ( z ) � φ ( T , z ) + φ ( t , x ) , ( t , x ) 2 [ 0 , T ) � R N . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Linearized formulation R � [ t , T ] , R N , U , dz � h � ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , dual formulation: η � ( t , x ) = 8 9 � [ 0 , T ] � R N � η 2 R : 9 φ 2 C 1 , 2 < s.t. = b 8 ( s , y , v , z ) 2 [ t , T ] � R N � V � R N , sup ; , : η � ( T � t ) L v φ ( s , y ) + h ( z ) � φ ( T , z ) + φ ( t , x ) , ( t , x ) 2 [ 0 , T ) � R N . In in…nite horizon (discounted) setting : Buckdahn, G., Quincampoix (preprint) Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Linearized formulation R � [ t , T ] , R N , U , dz � h � ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , dual formulation: η � ( t , x ) = 8 9 � [ 0 , T ] � R N � η 2 R : 9 φ 2 C 1 , 2 < s.t. = b 8 ( s , y , v , z ) 2 [ t , T ] � R N � V � R N , sup ; , : η � ( T � t ) L v φ ( s , y ) + h ( z ) � φ ( T , z ) + φ ( t , x ) , ( t , x ) 2 [ 0 , T ) � R N . In in…nite horizon (discounted) setting : Buckdahn, G., Quincampoix (preprint) Theorem ) V h ( t , x ) = h � ( t , x ) = η � ( t , x ) , h Lipschitz, bounded = 8 ( t , x ) 2 [ 0 , T ) � R N . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof ) V h ( t , x ) � h � ( t , x ) γ t , x , u 2 Θ ( t , x ) = Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof ) V h ( t , x ) � h � ( t , x ) γ t , x , u 2 Θ ( t , x ) = η � ( T � t ) L v φ ( s , y ) + h ( z ) � φ ( T , z ) + φ ( t , x ) integrate ) h � ( t , x ) � η � ( t , x ) . w.r.t. γ 2 Θ ( t , x ) = Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof ) V h ( t , x ) � h � ( t , x ) γ t , x , u 2 Θ ( t , x ) = η � ( T � t ) L v φ ( s , y ) + h ( z ) � φ ( T , z ) + φ ( t , x ) integrate ) h � ( t , x ) � η � ( t , x ) . w.r.t. γ 2 Θ ( t , x ) = approximate V h by smooth subsolutions V ε ; V ε ( t , x ) � C ε � η � ( t , x ) then ε ! 0 to get η � ( t , x ) � V h ( t , x ) Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Lower semicontinuous case R � [ t , T ] , R N , U , dz � V h ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , ( t , x ) 2 [ 0 , T ) � R N , V h ( T , � ) = h ( � ) . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Lower semicontinuous case R � [ t , T ] , R N , U , dz � V h ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , ( t , x ) 2 [ 0 , T ) � R N , V h ( T , � ) = h ( � ) . h : R N � ! R is a lower semicontinuous function. Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Lower semicontinuous case R � [ t , T ] , R N , U , dz � V h ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , ( t , x ) 2 [ 0 , T ) � R N , V h ( T , � ) = h ( � ) . h : R N � ! R is a lower semicontinuous function. � � j x j 2 + 1 9 c 2 R such that c � h ( x ) � � c , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Lower semicontinuous case R � [ t , T ] , R N , U , dz � V h ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , ( t , x ) 2 [ 0 , T ) � R N , V h ( T , � ) = h ( � ) . h : R N � ! R is a lower semicontinuous function. � � j x j 2 + 1 9 c 2 R such that c � h ( x ) � � c , Theorem V h is the smallest lower semicontinuous viscosity supersolution and V h ( t , x ) = η � ( t , x ) , 8 ( t , x ) 2 [ 0 , T ) � R N . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof inf-convolution h n ( x ) = inf y 2 R N ( h ( y ) ^ n + n j y � x j ) , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof inf-convolution h n ( x ) = inf y 2 R N ( h ( y ) ^ n + n j y � x j ) , h n ( x ) % h ( x ) , 8 x 2 R N . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof inf-convolution h n ( x ) = inf y 2 R N ( h ( y ) ^ n + n j y � x j ) , h n ( x ) % h ( x ) , 8 x 2 R N . R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof inf-convolution h n ( x ) = inf y 2 R N ( h ( y ) ^ n + n j y � x j ) , h n ( x ) % h ( x ) , 8 x 2 R N . R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ Θ ( t , x ) compact: V n ( t , x ) = R R N h n ( z ) γ n � [ t , T ] , R N , U , dz � Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof inf-convolution h n ( x ) = inf y 2 R N ( h ( y ) ^ n + n j y � x j ) , h n ( x ) % h ( x ) , 8 x 2 R N . R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ Θ ( t , x ) compact: V n ( t , x ) = R R N h n ( z ) γ n � [ t , T ] , R N , U , dz � V n ( t , x ) = 8 9 � [ 0 , T ] � R N � η 2 R : 9 φ 2 C 1 , 2 < s.t. = b 8 ( s , y , v , z ) 2 [ t , T ] � R N � V � R N , sup : ; η � ( T � t ) L v φ ( s , y ) + ( h n ( z ) � φ ( T , z )) + φ ( t , x ) . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof inf-convolution h n ( x ) = inf y 2 R N ( h ( y ) ^ n + n j y � x j ) , h n ( x ) % h ( x ) , 8 x 2 R N . R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ Θ ( t , x ) compact: V n ( t , x ) = R R N h n ( z ) γ n � [ t , T ] , R N , U , dz � V n ( t , x ) = 8 9 � [ 0 , T ] � R N � η 2 R : 9 φ 2 C 1 , 2 < s.t. = b 8 ( s , y , v , z ) 2 [ t , T ] � R N � V � R N , sup : ; η � ( T � t ) L v φ ( s , y ) + ( h n ( z ) � φ ( T , z )) + φ ( t , x ) . V n ( t , x ) � η � ( t , x ) � V h ( t , x ) Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof inf-convolution h n ( x ) = inf y 2 R N ( h ( y ) ^ n + n j y � x j ) , h n ( x ) % h ( x ) , 8 x 2 R N . R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ Θ ( t , x ) compact: V n ( t , x ) = R R N h n ( z ) γ n � [ t , T ] , R N , U , dz � V n ( t , x ) = 8 9 � [ 0 , T ] � R N � η 2 R : 9 φ 2 C 1 , 2 < s.t. = b 8 ( s , y , v , z ) 2 [ t , T ] � R N � V � R N , sup : ; η � ( T � t ) L v φ ( s , y ) + ( h n ( z ) � φ ( T , z )) + φ ( t , x ) . V n ( t , x ) � η � ( t , x ) � V h ( t , x ) W = sup n V n is the smallest l.s.c. supersolution Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof inf-convolution h n ( x ) = inf y 2 R N ( h ( y ) ^ n + n j y � x j ) , h n ( x ) % h ( x ) , 8 x 2 R N . R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ Θ ( t , x ) compact: V n ( t , x ) = R R N h n ( z ) γ n � [ t , T ] , R N , U , dz � V n ( t , x ) = 8 9 � [ 0 , T ] � R N � η 2 R : 9 φ 2 C 1 , 2 < s.t. = b 8 ( s , y , v , z ) 2 [ t , T ] � R N � V � R N , sup : ; η � ( T � t ) L v φ ( s , y ) + ( h n ( z ) � φ ( T , z )) + φ ( t , x ) . V n ( t , x ) � η � ( t , x ) � V h ( t , x ) W = sup n V n is the smallest l.s.c. supersolution m � n , V m ( t , x ) � R R N h n ( z ) γ m � [ t , T ] , R N , U , dz � ; m ! ∞ , n ! ∞ . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Upper semicontinuous case R � [ t , T ] , R N , U , dz � V h ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , ( t , x ) 2 [ 0 , T ) � R N , V h ( T , � ) = h ( � ) . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Upper semicontinuous case R � [ t , T ] , R N , U , dz � V h ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , ( t , x ) 2 [ 0 , T ) � R N , V h ( T , � ) = h ( � ) . h : R N � ! R is an upper semicontinuous function. Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Upper semicontinuous case R � [ t , T ] , R N , U , dz � V h ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , ( t , x ) 2 [ 0 , T ) � R N , V h ( T , � ) = h ( � ) . h : R N � ! R is an upper semicontinuous function. � � j x j 2 + 1 9 c 2 R such that - c � h ( x ) � c , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Upper semicontinuous case R � [ t , T ] , R N , U , dz � V h ( t , x ) = inf γ 2 Θ ( t , x ) R N h ( z ) γ , ( t , x ) 2 [ 0 , T ) � R N , V h ( T , � ) = h ( � ) . h : R N � ! R is an upper semicontinuous function. � � j x j 2 + 1 9 c 2 R such that - c � h ( x ) � c , Theorem V h is the largest upper semicontinuous viscosity subsolution of (HJB). Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof sup-convolution h n ( x ) = sup y 2 R N ( h ( y ) _ ( � n ) � n j y � x j ) Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof sup-convolution h n ( x ) = sup y 2 R N ( h ( y ) _ ( � n ) � n j y � x j ) � � �� V n ( t , x ) = inf u 2U E X t , x , u , ( t , x ) 2 [ 0 , T ) � R N . h n T Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof sup-convolution h n ( x ) = sup y 2 R N ( h ( y ) _ ( � n ) � n j y � x j ) � � �� V n ( t , x ) = inf u 2U E X t , x , u , ( t , x ) 2 [ 0 , T ) � R N . h n T R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof sup-convolution h n ( x ) = sup y 2 R N ( h ( y ) _ ( � n ) � n j y � x j ) � � �� V n ( t , x ) = inf u 2U E X t , x , u , ( t , x ) 2 [ 0 , T ) � R N . h n T R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ , W = inf n V n � V h . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Idea of the proof sup-convolution h n ( x ) = sup y 2 R N ( h ( y ) _ ( � n ) � n j y � x j ) � � �� V n ( t , x ) = inf u 2U E X t , x , u , ( t , x ) 2 [ 0 , T ) � R N . h n T R � [ t , T ] , R N , U , dz � V n ( t , x ) = inf γ 2 Θ ( t , x ) R N h n ( z ) γ , W = inf n V n � V h . γ 2 Θ ( t , x ) , V n ( t , x ) � R � [ t , T ] , R N , U , dz � R N h n ( z ) γ , pass n ! ∞ . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation What about the dual formulation? 1 � dX t , x = 0 , for 0 � t � s � T = 1 , s , h ( � ) = 1 f 0 g ( � ) . X t , x = x 2 R . t Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation What about the dual formulation? 1 � dX t , x = 0 , for 0 � t � s � T = 1 , s , h ( � ) = 1 f 0 g ( � ) . X t , x = x 2 R . t V h is the largest u.s.c. subsolution of � � ∂ t V h ( t , x ) = 0 , for all ( t , x ) 2 ( 0 , T ) � R , V h ( 1 , � ) = h ( � ) on R . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation What about the dual formulation? 1 � dX t , x = 0 , for 0 � t � s � T = 1 , s , h ( � ) = 1 f 0 g ( � ) . X t , x = x 2 R . t V h is the largest u.s.c. subsolution of � � ∂ t V h ( t , x ) = 0 , for all ( t , x ) 2 ( 0 , T ) � R , V h ( 1 , � ) = h ( � ) on R . V h ( t , � ) = h ( � ) , for every t 2 ( 0 , T ] Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation What about the dual formulation? 2 � 1 � = 1 . In particular, V h 2 , 0 Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation What about the dual formulation? 2 � 1 � = 1 . In particular, V h 2 , 0 η � � 1 � 2 , 0 8 9 η 2 R : 9 φ 2 C 1 , 2 < ([ 0 , 1 ] � R ) = � 1 � � R 2 , b = sup s.t. 8 ( s , y , z ) 2 2 , 1 � 1 � : ; η � 1 2 ∂ t φ ( s , y ) + h ( z ) � φ ( 1 , z ) + φ 2 , 0 Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation What about the dual formulation? 2 � 1 � = 1 . In particular, V h 2 , 0 η � � 1 � 2 , 0 8 9 η 2 R : 9 φ 2 C 1 , 2 < ([ 0 , 1 ] � R ) = � 1 � � R 2 , b = sup s.t. 8 ( s , y , z ) 2 2 , 1 � 1 � : ; η � 1 2 ∂ t φ ( s , y ) + h ( z ) � φ ( 1 , z ) + φ 2 , 0 z = ε , ε ! 0 to get η � � 1 � � 0 < V h � 1 � 2 , 0 2 , 0 . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. U.s.c. case � � π = Ω , F , ( F t ) t � 0 , P , W , u , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. U.s.c. case � � π = Ω , F , ( F t ) t � 0 , P , W , u , U w , Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. U.s.c. case � � π = Ω , F , ( F t ) t � 0 , P , W , u , U w , h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h . T Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. U.s.c. case � � π = Ω , F , ( F t ) t � 0 , P , W , u , U w , h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h . T Proposition If h is u.s.c., then V h ( t , x ) = V w h ( t , x ) , ( t , x ) 2 [ 0 , T ] � R N . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. U.s.c. case � � π = Ω , F , ( F t ) t � 0 , P , W , u , U w , h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h . T Proposition If h is u.s.c., then V h ( t , x ) = V w h ( t , x ) , ( t , x ) 2 [ 0 , T ] � R N . Idea of the proof : γ t , x , π ( A � B � C � D ) T � t E π h R T i P π � � 1 t 1 A � B � C (( s , X t , x , u X t , x , u = , u s )) ds 2 D s T Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. U.s.c. case � � π = Ω , F , ( F t ) t � 0 , P , W , u , U w , h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h . T Proposition If h is u.s.c., then V h ( t , x ) = V w h ( t , x ) , ( t , x ) 2 [ 0 , T ] � R N . Idea of the proof : γ t , x , π ( A � B � C � D ) T � t E π h R T i P π � � 1 t 1 A � B � C (( s , X t , x , u X t , x , u = , u s )) ds 2 D s T V n ( t , x ) = inf π 2U w E π � � �� � X t , x , u h n inf π 2U w E π � � �� � V h ( t , x ) T X t , x , u h T Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. L.s.c. case h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h T Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. L.s.c. case h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h T f σσ � ( t , x , u ) , b ( t , x , u ) : u 2 U g is convex. Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. L.s.c. case h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h T f σσ � ( t , x , u ) , b ( t , x , u ) : u 2 U g is convex. Proposition If convexity and h is l.s.c., then V h ( t , x ) = V w h ( t , x ) . Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. L.s.c. case h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h T f σσ � ( t , x , u ) , b ( t , x , u ) : u 2 U g is convex. Proposition If convexity and h is l.s.c., then V h ( t , x ) = V w h ( t , x ) . Idea of the proof : use inf-convolution, Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. L.s.c. case h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h T f σσ � ( t , x , u ) , b ( t , x , u ) : u 2 U g is convex. Proposition If convexity and h is l.s.c., then V h ( t , x ) = V w h ( t , x ) . Idea of the proof : use inf-convolution, V n ( t , x ) = inf π 2U w E π � � �� = X t , x , u h n R T X h n ( y T ) R n ( dydq ) , e � R + ; R N � � V , V is the for some control rule R n on e X = C set of positive Radon measures on R + � U whose projection on R + is the Lebesque measure. Dan Goreac On Some Discontinuous Control Problems

Linear formulation in Lipschitz case Deterministic framework Discontinuous setting Stochastic framework Weak control formulation Weak control formulation. L.s.c. case h ( t , x ) = inf π 2U w E π � � �� X t , x , u V w h T f σσ � ( t , x , u ) , b ( t , x , u ) : u 2 U g is convex. Proposition If convexity and h is l.s.c., then V h ( t , x ) = V w h ( t , x ) . Idea of the proof : use inf-convolution, V n ( t , x ) = inf π 2U w E π � � �� = X t , x , u h n R T X h n ( y T ) R n ( dydq ) , e � R + ; R N � � V , V is the for some control rule R n on e X = C set of positive Radon measures on R + � U whose projection on R + is the Lebesque measure. V m ( t , x ) � R X h n ( y T ) R m ( dydq ) if m � n ; e Dan Goreac On Some Discontinuous Control Problems